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# -*- coding: utf-8 -*- 

r""" 

Permutations 

 

The Permutations module. Use ``Permutation?`` to get information about 

the Permutation class, and ``Permutations?`` to get information about 

the combinatorial class of permutations. 

 

.. WARNING:: 

 

This file defined :class:`Permutation` which depends upon 

:class:`CombinatorialElement` despite it being deprecated (see 

:trac:`13742`). This is dangerous. In particular, the 

:meth:`Permutation._left_to_right_multiply_on_right` method (which can 

be called through multiplication) disables the input checks (see 

:meth:`Permutation`). This should not happen. Do not trust the results. 

 

What does this file define ? 

^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 

 

The main part of this file consists in the definition of permutation objects, 

i.e. the :meth:`Permutation` method and the 

:class:`~sage.combinat.permutation.Permutation` class. Global options for 

elements of the permutation class can be set through the 

:meth:`Permutations.options` object. 

 

Below are listed all methods and classes defined in this file. 

 

**Methods of Permutations objects** 

 

.. csv-table:: 

:class: contentstable 

:widths: 30, 70 

:delim: | 

 

:meth:`~sage.combinat.permutation.Permutation.left_action_product` | Returns the product of ``self`` with another permutation, in which the other permutation is applied first. 

:meth:`~sage.combinat.permutation.Permutation.right_action_product` | Returns the product of ``self`` with another permutation, in which ``self`` is applied first. 

:meth:`~sage.combinat.permutation.Permutation.size` | Returns the size of the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.cycle_string` | Returns the disjoint-cycles representation of ``self`` as string. 

:meth:`~sage.combinat.permutation.Permutation.next` | Returns the permutation that follows ``self`` in lexicographic order (in the same symmetric group as ``self``). 

:meth:`~sage.combinat.permutation.Permutation.prev` | Returns the permutation that comes directly before ``self`` in lexicographic order (in the same symmetric group as ``self``). 

:meth:`~sage.combinat.permutation.Permutation.to_tableau_by_shape` | Returns a tableau of shape ``shape`` with the entries in ``self``. 

:meth:`~sage.combinat.permutation.Permutation.to_cycles` | Returns the permutation ``self`` as a list of disjoint cycles. 

:meth:`~sage.combinat.permutation.Permutation.forget_cycles` | Return ``self`` under the forget cycle map. 

:meth:`~sage.combinat.permutation.Permutation.to_permutation_group_element` | Returns a ``PermutationGroupElement`` equal to ``self``. 

:meth:`~sage.combinat.permutation.Permutation.signature` | Returns the signature of the permutation ``sef``. 

:meth:`~sage.combinat.permutation.Permutation.is_even` | Returns ``True`` if the permutation ``self`` is even, and ``False`` otherwise. 

:meth:`~sage.combinat.permutation.Permutation.to_matrix` | Returns a matrix representing the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.rank` | Returns the rank of ``self`` in lexicographic ordering (on the symmetric group containing ``self``). 

:meth:`~sage.combinat.permutation.Permutation.to_inversion_vector` | Returns the inversion vector of a permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.inversions` | Returns a list of the inversions of permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.show` | Displays the permutation as a drawing. 

:meth:`~sage.combinat.permutation.Permutation.number_of_inversions` | Returns the number of inversions in the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.noninversions` | Returns the ``k``-noninversions in the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.number_of_noninversions` | Returns the number of ``k``-noninversions in the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.length` | Returns the Coxeter length of a permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.inverse` | Returns the inverse of a permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.ishift` | Returns the ``i``-shift of ``self``. 

:meth:`~sage.combinat.permutation.Permutation.iswitch` | Returns the ``i``-switch of ``self``. 

:meth:`~sage.combinat.permutation.Permutation.runs` | Returns a list of the runs in the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.longest_increasing_subsequence_length` | Returns the length of the longest increasing subsequences of ``self``. 

:meth:`~sage.combinat.permutation.Permutation.longest_increasing_subsequences` | Returns the list of the longest increasing subsequences of ``self``. 

:meth:`~sage.combinat.permutation.Permutation.cycle_type` | Returns the cycle type of ``self`` as a partition of ``len(self)``. 

:meth:`~sage.combinat.permutation.Permutation.foata_bijection` | Returns the image of the permutation ``self`` under the Foata bijection `\phi`. 

:meth:`~sage.combinat.permutation.Permutation.destandardize` | Return destandardization of ``self`` with respect to ``weight`` and ``ordered_alphabet``. 

:meth:`~sage.combinat.permutation.Permutation.to_lehmer_code` | Returns the Lehmer code of the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.to_lehmer_cocode` | Returns the Lehmer cocode of ``self``. 

:meth:`~sage.combinat.permutation.Permutation.reduced_word` | Returns the reduced word of the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.reduced_words` | Returns a list of the reduced words of the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.reduced_words_iterator` | An iterator for the reduced words of the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.reduced_word_lexmin` | Returns a lexicographically minimal reduced word of a permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.fixed_points` | Returns a list of the fixed points of the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.number_of_fixed_points` | Returns the number of fixed points of the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.recoils` | Returns the list of the positions of the recoils of the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.number_of_recoils` | Returns the number of recoils of the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.recoils_composition` | Returns the composition corresponding to the recoils of ``self``. 

:meth:`~sage.combinat.permutation.Permutation.descents` | Returns the list of the descents of the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.idescents` | Returns a list of the idescents of ``self``. 

:meth:`~sage.combinat.permutation.Permutation.idescents_signature` | Returns the list obtained by mapping each position in ``self`` to `-1` if it is an idescent and `1` if it is not an idescent. 

:meth:`~sage.combinat.permutation.Permutation.number_of_descents` | Returns the number of descents of the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.number_of_idescents` | Returns the number of idescents of the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.descents_composition` | Returns the composition corresponding to the descents of ``self``. 

:meth:`~sage.combinat.permutation.Permutation.descent_polynomial` | Returns the descent polynomial of the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.major_index` | Returns the major index of the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.imajor_index` | Returns the inverse major index of the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.to_major_code` | Returns the major code of the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.peaks` | Returns a list of the peaks of the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.number_of_peaks` | Returns the number of peaks of the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.saliances` | Returns a list of the saliances of the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.number_of_saliances` | Returns the number of saliances of the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.bruhat_lequal` | Returns ``True`` if self is less or equal to ``p2`` in the Bruhat order. 

:meth:`~sage.combinat.permutation.Permutation.weak_excedences` | Returns all the numbers ``self[i]`` such that ``self[i] >= i+1``. 

:meth:`~sage.combinat.permutation.Permutation.bruhat_inversions` | Returns the list of inversions of ``self`` such that the application of this inversion to ``self`` decrements its number of inversions. 

:meth:`~sage.combinat.permutation.Permutation.bruhat_inversions_iterator` | Returns an iterator over Bruhat inversions of ``self``. 

:meth:`~sage.combinat.permutation.Permutation.bruhat_succ` | Returns a list of the permutations covering ``self`` in the Bruhat order. 

:meth:`~sage.combinat.permutation.Permutation.bruhat_succ_iterator` | An iterator for the permutations covering ``self`` in the Bruhat order. 

:meth:`~sage.combinat.permutation.Permutation.bruhat_pred` | Returns a list of the permutations covered by ``self`` in the Bruhat order. 

:meth:`~sage.combinat.permutation.Permutation.bruhat_pred_iterator` | An iterator for the permutations covered by ``self`` in the Bruhat order. 

:meth:`~sage.combinat.permutation.Permutation.bruhat_smaller` | Returns the combinatorial class of permutations smaller than or equal to ``self`` in the Bruhat order. 

:meth:`~sage.combinat.permutation.Permutation.bruhat_greater` | Returns the combinatorial class of permutations greater than or equal to ``self`` in the Bruhat order. 

:meth:`~sage.combinat.permutation.Permutation.permutohedron_lequal` | Returns ``True`` if ``self`` is less or equal to ``p2`` in the permutohedron order. 

:meth:`~sage.combinat.permutation.Permutation.permutohedron_succ` | Returns a list of the permutations covering ``self`` in the permutohedron order. 

:meth:`~sage.combinat.permutation.Permutation.permutohedron_pred` | Returns a list of the permutations covered by ``self`` in the permutohedron order. 

:meth:`~sage.combinat.permutation.Permutation.permutohedron_smaller` | Returns a list of permutations smaller than or equal to ``self`` in the permutohedron order. 

:meth:`~sage.combinat.permutation.Permutation.permutohedron_greater` | Returns a list of permutations greater than or equal to ``self`` in the permutohedron order. 

:meth:`~sage.combinat.permutation.Permutation.right_permutohedron_interval_iterator` | Returns an iterator over permutations in an interval of the permutohedron order. 

:meth:`~sage.combinat.permutation.Permutation.right_permutohedron_interval` | Returns a list of permutations in an interval of the permutohedron order. 

:meth:`~sage.combinat.permutation.Permutation.has_pattern` | Tests whether the permutation ``self`` matches the pattern. 

:meth:`~sage.combinat.permutation.Permutation.avoids` | Tests whether the permutation ``self`` avoids the pattern. 

:meth:`~sage.combinat.permutation.Permutation.pattern_positions` | Returns the list of positions where the pattern ``patt`` appears in ``self``. 

:meth:`~sage.combinat.permutation.Permutation.reverse` | Returns the permutation obtained by reversing the 1-line notation of ``self``. 

:meth:`~sage.combinat.permutation.Permutation.complement` | Returns the complement of the permutation which is obtained by replacing each value `x` in the 1-line notation of ``self`` with `n - x + 1`. 

:meth:`~sage.combinat.permutation.Permutation.permutation_poset` | Returns the permutation poset of ``self``. 

:meth:`~sage.combinat.permutation.Permutation.dict` | Returns a dictionary corresponding to the permutation ``self``. 

:meth:`~sage.combinat.permutation.Permutation.action` | Returns the action of the permutation ``self`` on a list. 

:meth:`~sage.combinat.permutation.Permutation.robinson_schensted` | Returns the pair of standard tableaux obtained by running the Robinson-Schensted Algorithm on ``self``. 

:meth:`~sage.combinat.permutation.Permutation.left_tableau` | Returns the left standard tableau after performing the RSK algorithm. 

:meth:`~sage.combinat.permutation.Permutation.right_tableau` | Returns the right standard tableau after performing the RSK algorithm. 

:meth:`~sage.combinat.permutation.Permutation.increasing_tree` | Returns the increasing tree of ``self``. 

:meth:`~sage.combinat.permutation.Permutation.increasing_tree_shape` | Returns the shape of the increasing tree of ``self``. 

:meth:`~sage.combinat.permutation.Permutation.binary_search_tree` | Returns the binary search tree of ``self``. 

:meth:`~sage.combinat.permutation.Permutation.sylvester_class` | Iterates over the equivalence class of ``self`` under sylvester congruence 

:meth:`~sage.combinat.permutation.Permutation.RS_partition` | Returns the shape of the tableaux obtained by the RSK algorithm. 

:meth:`~sage.combinat.permutation.Permutation.remove_extra_fixed_points` | Returns the permutation obtained by removing any fixed points at the end of ``self``. 

:meth:`~sage.combinat.permutation.Permutation.retract_plain` | Returns the plain retract of ``self`` to a smaller symmetric group `S_m`. 

:meth:`~sage.combinat.permutation.Permutation.retract_direct_product` | Returns the direct-product retract of ``self`` to a smaller symmetric group `S_m`. 

:meth:`~sage.combinat.permutation.Permutation.retract_okounkov_vershik` | Returns the Okounkov-Vershik retract of ``self`` to a smaller symmetric group `S_m`. 

:meth:`~sage.combinat.permutation.Permutation.hyperoctahedral_double_coset_type` | Returns the coset-type of ``self`` as a partition. 

:meth:`~sage.combinat.permutation.Permutation.binary_search_tree_shape` | Returns the shape of the binary search tree of ``self`` (a non labelled binary tree). 

:meth:`~sage.combinat.permutation.Permutation.shifted_concatenation` | Returns the right (or left) shifted concatenation of ``self`` with a permutation ``other``. 

:meth:`~sage.combinat.permutation.Permutation.shifted_shuffle` | Returns the shifted shuffle of ``self`` with a permutation ``other``. 

 

**Other classes defined in this file** 

 

.. csv-table:: 

:class: contentstable 

:widths: 30, 70 

:delim: | 

 

:class:`Permutations` | 

:class:`Permutations_nk` | 

:class:`Permutations_mset` | 

:class:`Permutations_set` | 

:class:`Permutations_msetk` | 

:class:`Permutations_setk` | 

:class:`Arrangements` | 

:class:`Arrangements_msetk` | 

:class:`Arrangements_setk` | 

:class:`StandardPermutations_all` | 

:class:`StandardPermutations_n_abstract` | 

:class:`StandardPermutations_n` | 

:class:`StandardPermutations_descents` | 

:class:`StandardPermutations_recoilsfiner` | 

:class:`StandardPermutations_recoilsfatter` | 

:class:`StandardPermutations_recoils` | 

:class:`StandardPermutations_bruhat_smaller` | 

:class:`StandardPermutations_bruhat_greater` | 

:class:`CyclicPermutations` | 

:class:`CyclicPermutationsOfPartition` | 

:class:`StandardPermutations_avoiding_12` | 

:class:`StandardPermutations_avoiding_21` | 

:class:`StandardPermutations_avoiding_132` | 

:class:`StandardPermutations_avoiding_123` | 

:class:`StandardPermutations_avoiding_321` | 

:class:`StandardPermutations_avoiding_231` | 

:class:`StandardPermutations_avoiding_312` | 

:class:`StandardPermutations_avoiding_213` | 

:class:`StandardPermutations_avoiding_generic` | 

:class:`PatternAvoider` | 

 

**Functions defined in this file** 

 

.. csv-table:: 

:class: contentstable 

:widths: 30, 70 

:delim: | 

 

:meth:`from_major_code` | Returns the permutation corresponding to major code ``mc``. 

:meth:`from_permutation_group_element` | Returns a Permutation give a ``PermutationGroupElement`` ``pge``. 

:meth:`from_rank` | Returns the permutation with the specified lexicographic rank. 

:meth:`from_inversion_vector` | Returns the permutation corresponding to inversion vector ``iv``. 

:meth:`from_cycles` | Returns the permutation with given disjoint-cycle representation ``cycles``. 

:meth:`from_lehmer_code` | Returns the permutation with Lehmer code ``lehmer``. 

:meth:`from_reduced_word` | Returns the permutation corresponding to the reduced word ``rw``. 

:meth:`bistochastic_as_sum_of_permutations` | Returns a given bistochastic matrix as a nonnegative linear combination of permutations. 

:meth:`bounded_affine_permutation` | Returns a partial permutation representing the bounded affine permutation of a matrix. 

:meth:`descents_composition_list` | Returns a list of all the permutations in a given descent class (i. e., having a given descents composition). 

:meth:`descents_composition_first` | Returns the smallest element of a descent class. 

:meth:`descents_composition_last` | Returns the largest element of a descent class. 

:meth:`bruhat_lequal` | Returns ``True`` if ``p1`` is less or equal to ``p2`` in the Bruhat order. 

:meth:`permutohedron_lequal` | Returns ``True`` if ``p1`` is less or equal to ``p2`` in the permutohedron order. 

:meth:`to_standard` | Returns a standard permutation corresponding to the permutation ``self``. 

 

AUTHORS: 

 

- Mike Hansen 

 

- Dan Drake (2008-04-07): allow Permutation() to take lists of tuples 

 

- Sébastien Labbé (2009-03-17): added robinson_schensted_inverse 

 

- Travis Scrimshaw: 

 

* (2012-08-16): ``to_standard()`` no longer modifies input 

* (2013-01-19): Removed RSK implementation and moved to 

:mod:`~sage.combinat.rsk`. 

* (2013-07-13): Removed ``CombinatorialClass`` and moved permutations to the 

category framework. 

 

- Darij Grinberg (2013-09-07): added methods; ameliorated :trac:`14885` by 

exposing and documenting methods for global-independent 

multiplication. 

 

- Travis Scrimshaw (2014-02-05): Made :class:`StandardPermutations_n` a 

finite Weyl group to make it more uniform with :class:`SymmetricGroup`. 

Added ability to compute the conjugacy classes. 

 

Classes and methods 

=================== 

""" 

 

#***************************************************************************** 

# Copyright (C) 2007 Mike Hansen <mhansen@gmail.com> 

# 

# This program is free software: you can redistribute it and/or modify 

# it under the terms of the GNU General Public License as published by 

# the Free Software Foundation, either version 2 of the License, or 

# (at your option) any later version. 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from __future__ import print_function, absolute_import 

 

from builtins import zip 

from six.moves import range 

 

from six import itervalues 

 

from sage.structure.parent import Parent 

from sage.structure.unique_representation import UniqueRepresentation 

from sage.categories.infinite_enumerated_sets import InfiniteEnumeratedSets 

from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets 

from sage.categories.finite_weyl_groups import FiniteWeylGroups 

from sage.categories.finite_permutation_groups import FinitePermutationGroups 

from sage.structure.list_clone import ClonableArray 

from sage.structure.global_options import GlobalOptions 

from sage.interfaces.all import gap 

from sage.rings.all import ZZ, Integer, PolynomialRing 

from sage.arith.all import factorial 

from sage.matrix.matrix_space import MatrixSpace 

from sage.combinat.tools import transitive_ideal 

from sage.combinat.composition import Composition 

from sage.groups.perm_gps.permgroup_named import SymmetricGroup 

from sage.groups.perm_gps.permgroup_element import PermutationGroupElement 

from sage.misc.prandom import sample 

from sage.graphs.digraph import DiGraph 

import itertools 

from .combinat import CombinatorialElement, catalan_number 

from sage.misc.misc import uniq 

from sage.misc.cachefunc import cached_method 

from .backtrack import GenericBacktracker 

from sage.combinat.combinatorial_map import combinatorial_map 

from sage.combinat.rsk import RSK, RSK_inverse 

from sage.combinat.permutation_cython import (left_action_product, 

right_action_product, left_action_same_n, right_action_same_n, 

map_to_list, next_perm) 

 

class Permutation(CombinatorialElement): 

r""" 

A permutation. 

 

Converts ``l`` to a permutation on `\{1, 2, \ldots, n\}`. 

 

INPUT: 

 

- ``l`` -- Can be any one of the following: 

 

- an instance of :class:`Permutation`, 

 

- list of integers, viewed as one-line permutation notation. The 

construction checks that you give an acceptable entry. To avoid 

the check, use the ``check_input`` option. 

 

- string, expressing the permutation in cycle notation. 

 

- list of tuples of integers, expressing the permutation in cycle 

notation. 

 

- a :class:`PermutationGroupElement` 

 

- a pair of two standard tableaux of the same shape. This yields 

the permutation obtained from the pair using the inverse of the 

Robinson-Schensted algorithm. 

 

- ``check_input`` (boolean) -- whether to check that input is correct. Slows 

the function down, but ensures that nothing bad happens. This is set to 

``True`` by default. 

 

.. WARNING:: 

 

Since :trac:`13742` the input is checked for correctness : it is not 

accepted unless it actually is a permutation on `\{1, \ldots, n\}`. It 

means that some :meth:`Permutation` objects cannot be created anymore 

without setting ``check_input = False``, as there is no certainty that 

its functions can handle them, and this should be fixed in a much 

better way ASAP (the functions should be rewritten to handle those 

cases, and new tests be added). 

 

.. WARNING:: 

 

There are two possible conventions for multiplying permutations, and 

the one currently enabled in Sage by default is the one which has 

`(pq)(i) = q(p(i))` for any permutations `p \in S_n` and `q \in S_n` 

and any `1 \leq i \leq n`. (This equation looks less strange when 

the action of permutations on numbers is written from the right: 

then it takes the form `i^{pq} = (i^p)^q`, which is an associativity 

law). There is an alternative convention, which has 

`(pq)(i) = p(q(i))` instead. The conventions can be switched at 

runtime using 

:meth:`sage.combinat.permutation.Permutations.options`. 

It is best for code not to rely on this setting being set to a 

particular standard, but rather use the methods 

:meth:`left_action_product` and :meth:`right_action_product` for 

multiplying permutations (these methods don't depend on the setting). 

See :trac:`14885` for more details. 

 

.. NOTE:: 

 

The ``bruhat*`` methods refer to the *strong* Bruhat order. To use 

the *weak* Bruhat order, look under ``permutohedron*``. 

 

EXAMPLES:: 

 

sage: Permutation([2,1]) 

[2, 1] 

sage: Permutation([2, 1, 4, 5, 3]) 

[2, 1, 4, 5, 3] 

sage: Permutation('(1,2)') 

[2, 1] 

sage: Permutation('(1,2)(3,4,5)') 

[2, 1, 4, 5, 3] 

sage: Permutation( ((1,2),(3,4,5)) ) 

[2, 1, 4, 5, 3] 

sage: Permutation( [(1,2),(3,4,5)] ) 

[2, 1, 4, 5, 3] 

sage: Permutation( ((1,2)) ) 

[2, 1] 

sage: Permutation( (1,2) ) 

[2, 1] 

sage: Permutation( ((1,2),) ) 

[2, 1] 

sage: Permutation( ((1,),) ) 

[1] 

sage: Permutation( (1,) ) 

[1] 

sage: Permutation( () ) 

[] 

sage: Permutation( ((),) ) 

[] 

sage: p = Permutation((1, 2, 5)); p 

[2, 5, 3, 4, 1] 

sage: type(p) 

<class 'sage.combinat.permutation.StandardPermutations_n_with_category.element_class'> 

 

Construction from a string in cycle notation:: 

 

sage: p = Permutation( '(4,5)' ); p 

[1, 2, 3, 5, 4] 

 

The size of the permutation is the maximum integer appearing; add 

a 1-cycle to increase this:: 

 

sage: p2 = Permutation( '(4,5)(10)' ); p2 

[1, 2, 3, 5, 4, 6, 7, 8, 9, 10] 

sage: len(p); len(p2) 

5 

10 

 

We construct a :class:`Permutation` from a 

:class:`PermutationGroupElement`:: 

 

sage: g = PermutationGroupElement([2,1,3]) 

sage: Permutation(g) 

[2, 1, 3] 

 

From a pair of tableaux of the same shape. This uses the inverse 

of the Robinson-Schensted algorithm:: 

 

sage: p = [[1, 4, 7], [2, 5], [3], [6]] 

sage: q = [[1, 2, 5], [3, 6], [4], [7]] 

sage: P = Tableau(p) 

sage: Q = Tableau(q) 

sage: Permutation( (p, q) ) 

[3, 6, 5, 2, 7, 4, 1] 

sage: Permutation( [p, q] ) 

[3, 6, 5, 2, 7, 4, 1] 

sage: Permutation( (P, Q) ) 

[3, 6, 5, 2, 7, 4, 1] 

sage: Permutation( [P, Q] ) 

[3, 6, 5, 2, 7, 4, 1] 

 

TESTS:: 

 

sage: Permutation([()]) 

[] 

sage: Permutation('()') 

[] 

sage: Permutation(()) 

[] 

sage: Permutation( [1] ) 

[1] 

 

From a pair of empty tableaux :: 

 

sage: Permutation( ([], []) ) 

[] 

sage: Permutation( [[], []] ) 

[] 

 

.. automethod:: _left_to_right_multiply_on_right 

.. automethod:: _left_to_right_multiply_on_left 

""" 

@staticmethod 

def __classcall_private__(cls, l, check_input = True): 

""" 

Return a permutation in the general permutations parent. 

 

EXAMPLES:: 

 

sage: P = Permutation([2,1]); P 

[2, 1] 

sage: P.parent() 

Standard permutations 

""" 

import sage.combinat.tableau as tableau 

if isinstance(l, Permutation): 

return l 

elif isinstance(l, PermutationGroupElement): 

l = l.domain() 

#if l is a string, then assume it is in cycle notation 

elif isinstance(l, str): 

if l == "()" or l == "": 

return from_cycles(0, []) 

cycles = l.split(")(") 

cycles[0] = cycles[0][1:] 

cycles[-1] = cycles[-1][:-1] 

cycle_list = [] 

for c in cycles: 

cycle_list.append([int(_) for _ in c.split(",")]) 

 

return from_cycles(max(max(c) for c in cycle_list), cycle_list) 

 

#if l is a pair of standard tableaux or a pair of lists 

elif isinstance(l, (tuple, list)) and len(l) == 2 and \ 

all(isinstance(x, tableau.Tableau) for x in l): 

return RSK_inverse(*l, output='permutation') 

elif isinstance(l, (tuple, list)) and len(l) == 2 and \ 

all(isinstance(x, list) for x in l): 

P,Q = [tableau.Tableau(_) for _ in l] 

return RSK_inverse(P, Q, 'permutation') 

# if it's a tuple or nonempty list of tuples, also assume cycle 

# notation 

elif isinstance(l, tuple) or \ 

(isinstance(l, list) and l and 

all(isinstance(x, tuple) for x in l)): 

if l and (isinstance(l[0], (int,Integer)) or len(l[0]) > 0): 

if isinstance(l[0], tuple): 

n = max(max(x) for x in l) 

return from_cycles(n, [list(x) for x in l]) 

else: 

n = max(l) 

return from_cycles(n, [list(l)]) 

elif len(l) <= 1: 

return Permutations()([]) 

else: 

raise ValueError("cannot convert l (= %s) to a Permutation"%l) 

 

# otherwise, it gets processed by CombinatorialElement's __init__. 

return Permutations()(l, check_input=check_input) 

 

def __init__(self, parent, l, check_input=True): 

""" 

Constructor. Checks that INPUT is not a mess, and calls 

:class:`CombinatorialElement`. It should not, because 

:class:`CombinatorialElement` is deprecated. 

 

INPUT: 

 

- ``l`` -- a list of ``int`` variables 

 

- ``check_input`` (boolean) -- whether to check that input is 

correct. Slows the function down, but ensures that nothing bad 

happens. 

 

This is set to ``True`` by default. 

 

TESTS:: 

 

sage: Permutation([1,2,3]) 

[1, 2, 3] 

sage: Permutation([1,2,2,4]) 

Traceback (most recent call last): 

... 

ValueError: An element appears twice in the input. It should not. 

sage: Permutation([1,2,4,-1]) 

Traceback (most recent call last): 

... 

ValueError: the elements must be strictly positive integers 

sage: Permutation([1,2,4,5]) 

Traceback (most recent call last): 

... 

ValueError: The permutation has length 4 but its maximal element is 

5. Some element may be repeated, or an element is missing, but there 

is something wrong with its length. 

""" 

l = list(l) 

 

if check_input and len(l) > 0: 

# Make a copy to sort later 

lst = list(l) 

 

# Is input a list of positive integers ? 

for i in lst: 

try: 

i = int(i) 

except TypeError: 

raise ValueError("the elements must be integer variables") 

if i < 1: 

raise ValueError("the elements must be strictly positive integers") 

 

lst.sort() 

 

# Is the maximum element of the permutation the length of input, 

# or is some integer missing ? 

if int(lst[-1]) != len(lst): 

raise ValueError("The permutation has length "+str(len(lst))+ 

" but its maximal element is "+ 

str(int(lst[-1]))+". Some element "+ 

"may be repeated, or an element is missing"+ 

", but there is something wrong with its length.") 

 

# Do the elements appear only once ? 

previous = lst[0]-1 

 

for i in lst: 

if i == previous: 

raise ValueError("An element appears twice in the input. It should not.") 

previous = i 

 

CombinatorialElement.__init__(self, parent, l) 

 

def __setstate__(self, state): 

r""" 

In order to maintain backwards compatibility and be able to unpickle a 

old pickle from ``Permutation_class`` we have to override the default 

``__setstate__``. 

 

EXAMPLES:: 

 

sage: loads(b'x\x9ck`J.NLO\xd5K\xce\xcfM\xca\xccK,\xd1+H-\xca--I,\xc9\xcc\xcf\xe3\n@\xb0\xe3\x93s\x12\x8b\x8b\xb9\n\x195\x1b' 

....: b'\x0b\x99j\x0b\x995BY\xe33\x12\x8b3\nY\xfc\x80\xac\x9c\xcc\xe2\x92B\xd6\xd8B6\r\x88iE\x99y\xe9\xc5z\x99y%\xa9\xe9' 

....: b'\xa9E\\\xb9\x89\xd9\xa9\xf10N!{(\xa3qkP!G\x06\x90a\x04dp\x82\x18\x86@\x06Wji\x92\x1e\x00i\x8d0q') 

[3, 2, 1] 

sage: loads(dumps( Permutation([3,2,1]) )) # indirect doctest 

[3, 2, 1] 

""" 

if isinstance(state, dict): # for old pickles from Permutation_class 

self._set_parent(Permutations()) 

self.__dict__ = state 

else: 

self._set_parent(state[0]) 

self.__dict__ = state[1] 

 

@cached_method 

def __hash__(self): 

""" 

TESTS:: 

 

sage: d = {} 

sage: p = Permutation([1,2,3]) 

sage: d[p] = 1 

sage: d[p] 

1 

""" 

try: 

return hash(tuple(self._list)) 

except Exception: 

return hash(str(self._list)) 

 

def __str__(self): 

""" 

TESTS:: 

 

sage: Permutations.options.display='list' 

sage: p = Permutation([2,1,3]) 

sage: str(p) 

'[2, 1, 3]' 

sage: Permutations.options.display='cycle' 

sage: str(p) 

'(1,2)' 

sage: Permutations.options.display='singleton' 

sage: str(p) 

'(1,2)(3)' 

sage: Permutations.options.display='list' 

""" 

return repr(self) 

 

def _repr_(self): 

""" 

TESTS:: 

 

sage: p = Permutation([2,1,3]) 

sage: p 

[2, 1, 3] 

sage: Permutations.options.display='cycle' 

sage: p 

(1,2) 

sage: Permutations.options.display='singleton' 

sage: p 

(1,2)(3) 

sage: Permutations.options.display='reduced_word' 

sage: p 

[1] 

sage: Permutations.options._reset() 

""" 

display = self.parent().options.display 

if display == 'list': 

return repr(self._list) 

elif display == 'cycle': 

return self.cycle_string() 

elif display == 'singleton': 

return self.cycle_string(singletons=True) 

elif display == 'reduced_word': 

return repr(self.reduced_word()) 

 

def _latex_(self): 

r""" 

Return a `\LaTeX` representation of ``self``. 

 

EXAMPLES:: 

 

sage: p = Permutation([2,1,3]) 

sage: latex(p) 

[2, 1, 3] 

sage: Permutations.options.latex='cycle' 

sage: latex(p) 

(1 \; 2) 

sage: Permutations.options.latex='singleton' 

sage: latex(p) 

(1 \; 2)(3) 

sage: Permutations.options.latex='reduced_word' 

sage: latex(p) 

s_{1} 

sage: latex(Permutation([1,2,3])) 

1 

sage: Permutations.options.latex_empty_str="e" 

sage: latex(Permutation([1,2,3])) 

e 

sage: Permutations.options.latex='twoline' 

sage: latex(p) 

\begin{pmatrix} 1 & 2 & 3 \\ 2 & 1 & 3 \end{pmatrix} 

sage: Permutations.options._reset() 

""" 

display = self.parent().options.latex 

if display == "reduced_word": 

let = self.parent().options.generator_name 

redword = self.reduced_word() 

if not redword: 

return self.parent().options.latex_empty_str 

return " ".join("%s_{%s}"%(let, i) for i in redword) 

if display == "twoline": 

return "\\begin{pmatrix} %s \\\\ %s \\end{pmatrix}"%( 

" & ".join("%s"%i for i in range(1, len(self._list)+1)), 

" & ".join("%s"%i for i in self._list)) 

if display == "list": 

return repr(self._list) 

if display == "cycle": 

ret = self.cycle_string() 

else: # Must be cycles with singletons 

ret = self.cycle_string(singletons=True) 

return ret.replace(",", " \\; ") 

 

def _gap_(self, gap): 

""" 

Return a GAP version of this permutation. 

 

EXAMPLES:: 

 

sage: gap(Permutation([1,2,3])) 

() 

sage: gap(Permutation((1,2,3))) 

(1,2,3) 

sage: type(_) 

<class 'sage.interfaces.gap.GapElement'> 

""" 

return self.to_permutation_group_element()._gap_(gap) 

 

def size(self): 

""" 

Return the size of ``self``. 

 

EXAMPLES:: 

 

sage: Permutation([3,4,1,2,5]).size() 

5 

""" 

return len(self) 

 

def cycle_string(self, singletons=False): 

""" 

Returns a string of the permutation in cycle notation. 

 

If ``singletons=True``, it includes 1-cycles in the string. 

 

EXAMPLES:: 

 

sage: Permutation([1,2,3]).cycle_string() 

'()' 

sage: Permutation([2,1,3]).cycle_string() 

'(1,2)' 

sage: Permutation([2,3,1]).cycle_string() 

'(1,2,3)' 

sage: Permutation([2,1,3]).cycle_string(singletons=True) 

'(1,2)(3)' 

""" 

cycles = self.to_cycles(singletons=singletons) 

if cycles == []: 

return "()" 

else: 

return "".join(["("+",".join([str(l) for l in x])+")" for x in cycles]) 

 

def __next__(self): 

r""" 

Return the permutation that follows ``self`` in lexicographic order on 

the symmetric group containing ``self``. If ``self`` is the last 

permutation, then ``next`` returns ``False``. 

 

EXAMPLES:: 

 

sage: p = Permutation([1, 3, 2]) 

sage: next(p) 

[2, 1, 3] 

sage: p = Permutation([4,3,2,1]) 

sage: next(p) 

False 

 

TESTS:: 

 

sage: p = Permutation([]) 

sage: next(p) 

False 

""" 

p = self[:] 

n = len(self) 

first = -1 

 

#Starting from the end, find the first o such that 

#p[o] < p[o+1] 

for i in reversed(range(0,n-1)): 

if p[i] < p[i+1]: 

first = i 

break 

 

#If first is still -1, then we are already at the last permutation 

if first == -1: 

return False 

 

#Starting from the end, find the first j such that p[j] > p[first] 

j = n - 1 

while p[j] < p[first]: 

j -= 1 

 

#Swap positions first and j 

(p[j], p[first]) = (p[first], p[j]) 

 

#Reverse the list between first and the end 

first_half = p[:first+1] 

last_half = p[first+1:] 

last_half.reverse() 

p = first_half + last_half 

 

return Permutations()(p) 

 

next = __next__ 

 

def prev(self): 

r""" 

Return the permutation that comes directly before ``self`` in 

lexicographic order on the symmetric group containing ``self``. 

If ``self`` is the first permutation, then it returns ``False``. 

 

EXAMPLES:: 

 

sage: p = Permutation([1,2,3]) 

sage: p.prev() 

False 

sage: p = Permutation([1,3,2]) 

sage: p.prev() 

[1, 2, 3] 

 

TESTS:: 

 

sage: p = Permutation([]) 

sage: p.prev() 

False 

 

Check that :trac:`16913` is fixed:: 

 

sage: Permutation([1,4,3,2]).prev() 

[1, 4, 2, 3] 

""" 

 

p = self[:] 

n = len(self) 

first = -1 

 

#Starting from the end, find the first o such that 

#p[o] > p[o+1] 

for i in reversed(range(0, n-1)): 

if p[i] > p[i+1]: 

first = i 

break 

 

#If first is still -1, that is we didn't find any descents, 

#then we are already at the last permutation 

if first == -1: 

return False 

 

#Starting from the end, find the first j such that p[j] < p[first] 

j = n - 1 

while p[j] > p[first]: 

j -= 1 

 

#Swap positions first and j 

(p[j], p[first]) = (p[first], p[j]) 

 

#Reverse the list between first+1 and end 

first_half = p[:first+1] 

last_half = p[first+1:] 

last_half.reverse() 

p = first_half + last_half 

 

return Permutations()(p) 

 

 

def to_tableau_by_shape(self, shape): 

""" 

Return a tableau of shape ``shape`` with the entries 

in ``self``. The tableau is such that the reading word (i. e., 

the word obtained by reading the tableau row by row, starting 

from the top row in English notation, with each row being 

read from left to right) is ``self``. 

 

EXAMPLES:: 

 

sage: Permutation([3,4,1,2,5]).to_tableau_by_shape([3,2]) 

[[1, 2, 5], [3, 4]] 

sage: Permutation([3,4,1,2,5]).to_tableau_by_shape([3,2]).reading_word_permutation() 

[3, 4, 1, 2, 5] 

""" 

import sage.combinat.tableau as tableau 

if sum(shape) != len(self): 

raise ValueError("the size of the partition must be the size of self") 

 

t = [] 

w = list(self) 

for i in reversed(shape): 

t = [ w[:i] ] + t 

w = w[i:] 

return tableau.Tableau(t) 

 

def to_cycles(self, singletons=True): 

""" 

Return the permutation ``self`` as a list of disjoint cycles. 

 

The cycles are returned in the order of increasing smallest 

elements, and each cycle is returned as a tuple which starts 

with its smallest element. 

 

If ``singletons=False`` is given, the list does not contain the 

singleton cycles. 

 

EXAMPLES:: 

 

sage: Permutation([2,1,3,4]).to_cycles() 

[(1, 2), (3,), (4,)] 

sage: Permutation([2,1,3,4]).to_cycles(singletons=False) 

[(1, 2)] 

 

sage: Permutation([4,1,5,2,6,3]).to_cycles() 

[(1, 4, 2), (3, 5, 6)] 

 

The algorithm is of complexity `O(n)` where `n` is the size of the 

given permutation. 

 

TESTS:: 

 

sage: from sage.combinat.permutation import from_cycles 

sage: for n in range(1,6): 

....: for p in Permutations(n): 

....: if from_cycles(n, p.to_cycles()) != p: 

....: print("There is a problem with {}".format(p)) 

....: break 

sage: size = 10000 

sage: sample = (Permutations(size).random_element() for i in range(5)) 

sage: all(from_cycles(size, p.to_cycles()) == p for p in sample) 

True 

 

Note: there is an alternative implementation called ``_to_cycle_set`` 

which could be slightly (10%) faster for some input (typically for 

permutations of size in the range [100, 10000]). You can run the 

following benchmarks. For small permutations:: 

 

sage: for size in range(9): # not tested 

....: print(size) 

....: lp = Permutations(size).list() 

....: timeit('[p.to_cycles(False) for p in lp]') 

....: timeit('[p._to_cycles_set(False) for p in lp]') 

....: timeit('[p._to_cycles_list(False) for p in lp]') 

....: timeit('[p._to_cycles_orig(False) for p in lp]') 

 

and larger ones:: 

 

sage: for size in [10, 20, 50, 75, 100, 200, 500, 1000, # not tested 

....: 2000, 5000, 10000, 15000, 20000, 30000, 

....: 50000, 80000, 100000]: 

....: print(size) 

....: lp = [Permutations(size).random_element() for i in range(20)] 

....: timeit("[p.to_cycles() for p in lp]") 

....: timeit("[p._to_cycles_set() for p in lp]") 

....: timeit("[p._to_cycles_list() for p in lp]") 

""" 

cycles = [] 

 

l = self[:] 

 

# Go through until we've considered every number between 1 and len(l) 

for i in range(len(l)): 

if not l[i]: 

continue 

cycleFirst = i + 1 

cycle = [cycleFirst] 

l[i], next = False, l[i] 

while next != cycleFirst: 

cycle.append( next ) 

l[next - 1], next = False, l[next - 1] 

# Add the cycle to the list of cycles 

if singletons or len(cycle) > 1: 

cycles.append(tuple(cycle)) 

return cycles 

 

cycle_tuples = to_cycles 

 

def _to_cycles_orig(self, singletons=True): 

r""" 

Returns the permutation ``self`` as a list of disjoint cycles. 

 

EXAMPLES:: 

 

sage: Permutation([2,1,3,4])._to_cycles_orig() 

[(1, 2), (3,), (4,)] 

sage: Permutation([2,1,3,4])._to_cycles_orig(singletons=False) 

[(1, 2)] 

""" 

p = self[:] 

cycles = [] 

toConsider = -1 

 

#Create the list [1,2,...,len(p)] 

l = [ i+1 for i in range(len(p))] 

cycle = [] 

 

#Go through until we've considered every number between 

#1 and len(p) 

while len(l) > 0: 

#If we are at the end of a cycle 

#then we want to add it to the cycles list 

if toConsider == -1: 

#Add the cycle to the list of cycles 

if singletons: 

if cycle != []: 

cycles.append(tuple(cycle)) 

else: 

if len(cycle) > 1: 

cycles.append(tuple(cycle)) 

#Start with the first element in the list 

toConsider = l[0] 

l.remove(toConsider) 

cycle = [ toConsider ] 

cycleFirst = toConsider 

 

#Figure out where the element under consideration 

#gets mapped to. 

next = p[toConsider - 1] 

 

#If the next element is the first one in the list 

#then we've reached the end of the cycle 

if next == cycleFirst: 

toConsider = -1 

else: 

cycle.append( next ) 

l.remove( next ) 

toConsider = next 

 

#When we're finished, add the last cycle 

if singletons: 

if cycle != []: 

cycles.append(tuple(cycle)) 

else: 

if len(cycle) > 1: 

cycles.append(tuple(cycle)) 

return cycles 

 

def _to_cycles_set(self, singletons=True): 

r""" 

Return the permutation ``self`` as a list of disjoint cycles. 

 

EXAMPLES:: 

 

sage: Permutation([2,1,3,4])._to_cycles_set() 

[(1, 2), (3,), (4,)] 

sage: Permutation([2,1,3,4])._to_cycles_set(singletons=False) 

[(1, 2)] 

 

TESTS:: 

 

sage: all((p._to_cycles_set(False) == p._to_cycles_orig(False) 

....: for i in range(7) for p in Permutations(i))) 

True 

""" 

p = self[:] 

cycles = [] 

 

if not singletons: 

#remove the fixed points 

L = set( i+1 for i,pi in enumerate(p) if pi != i+1 ) 

else: 

L = set(range(1,len(p)+1)) 

 

#Go through until we've considered every remaining number 

while len(L) > 0: 

# take the first remaining element 

cycleFirst = L.pop() 

next = p[cycleFirst-1] 

cycle = [cycleFirst] 

while next != cycleFirst: 

cycle.append(next) 

L.remove(next) 

next = p[next-1] 

# add the cycle 

cycles.append(tuple(cycle)) 

 

return cycles 

 

def _to_cycles_list(self, singletons=True): 

r""" 

Return the permutation ``self`` as a list of disjoint cycles. 

 

EXAMPLES:: 

 

sage: Permutation([2,1,3,4])._to_cycles_list() 

[(1, 2), (3,), (4,)] 

sage: Permutation([2,1,3,4])._to_cycles_list(singletons=False) 

[(1, 2)] 

 

TESTS:: 

 

sage: all((p._to_cycles_list(False) == p._to_cycles_orig(False) 

....: for i in range(7) for p in Permutations(i))) 

True 

""" 

p = self[:] 

cycles = [] 

 

if not singletons: 

#remove the fixed points 

L = [i+1 for i,pi in enumerate(p) if pi != i+1] 

else: 

L = list(range(1, len(p) + 1)) 

 

from bisect import bisect_left 

 

#Go through until we've considered every remaining number 

while len(L) > 0: 

# take the first remaining element 

cycleFirst = L.pop(0) 

next = p[cycleFirst-1] 

cycle = [cycleFirst] 

while next != cycleFirst: 

cycle.append(next) 

# remove next from L 

# we use a binary search to find it 

L.pop(bisect_left(L,next)) 

next = p[next-1] 

# add the cycle 

cycles.append(tuple(cycle)) 

 

return cycles 

 

 

def to_permutation_group_element(self): 

""" 

Returns a PermutationGroupElement equal to self. 

 

EXAMPLES:: 

 

sage: Permutation([2,1,4,3]).to_permutation_group_element() 

(1,2)(3,4) 

sage: Permutation([1,2,3]).to_permutation_group_element() 

() 

""" 

cycles = self.to_cycles(singletons=False) 

grp = SymmetricGroup(len(self)) 

if cycles == []: 

return PermutationGroupElement( '()', parent=grp ) 

else: 

return PermutationGroupElement( cycles , parent=grp) 

 

def signature(self): 

r""" 

Return the signature of the permutation ``self``. This is 

`(-1)^l`, where `l` is the number of inversions of ``self``. 

 

.. NOTE:: 

 

:meth:`sign` can be used as an alias for :meth:`signature`. 

 

EXAMPLES:: 

 

sage: Permutation([4, 2, 3, 1, 5]).signature() 

-1 

sage: Permutation([1,3,2,5,4]).sign() 

1 

sage: Permutation([]).sign() 

1 

""" 

return (-1)**(len(self)-len(self.to_cycles())) 

 

#one can also use sign as an alias for signature 

sign = signature 

 

def is_even(self): 

r""" 

Return ``True`` if the permutation ``self`` is even and 

``False`` otherwise. 

 

EXAMPLES:: 

 

sage: Permutation([1,2,3]).is_even() 

True 

sage: Permutation([2,1,3]).is_even() 

False 

""" 

return self.signature() == 1 

 

def to_matrix(self): 

r""" 

Return a matrix representing the permutation. 

 

EXAMPLES:: 

 

sage: Permutation([1,2,3]).to_matrix() 

[1 0 0] 

[0 1 0] 

[0 0 1] 

 

Alternatively:: 

 

sage: matrix(Permutation([1,3,2])) 

[1 0 0] 

[0 0 1] 

[0 1 0] 

 

Notice that matrix multiplication corresponds to permutation 

multiplication only when the permutation option mult='r2l' 

 

:: 

 

sage: Permutations.options.mult='r2l' 

sage: p = Permutation([2,1,3]) 

sage: q = Permutation([3,1,2]) 

sage: (p*q).to_matrix() 

[0 0 1] 

[0 1 0] 

[1 0 0] 

sage: p.to_matrix()*q.to_matrix() 

[0 0 1] 

[0 1 0] 

[1 0 0] 

sage: Permutations.options.mult='l2r' 

sage: (p*q).to_matrix() 

[1 0 0] 

[0 0 1] 

[0 1 0] 

""" 

# build the dictionary of entries since the matrix is 

# extremely sparse 

entries = { (v-1, i): 1 for i, v in enumerate(self) } 

M = MatrixSpace(ZZ, len(self), sparse=True) 

return M(entries) 

 

_matrix_ = to_matrix 

 

@combinatorial_map(name='to alternating sign matrix') 

def to_alternating_sign_matrix(self): 

r""" 

Return a matrix representing the permutation in the 

:class:`AlternatingSignMatrix` class. 

 

EXAMPLES:: 

 

sage: m = Permutation([1,2,3]).to_alternating_sign_matrix(); m 

[1 0 0] 

[0 1 0] 

[0 0 1] 

sage: parent(m) 

Alternating sign matrices of size 3 

""" 

from sage.combinat.alternating_sign_matrix import AlternatingSignMatrix 

return AlternatingSignMatrix(self.to_matrix().rows()) 

 

def __mul__(self, rp): 

""" 

TESTS:: 

 

sage: p213 = Permutation([2,1,3]) 

sage: p312 = Permutation([3,1,2]) 

sage: Permutations.options.mult='l2r' 

sage: p213*p312 

[1, 3, 2] 

sage: Permutations.options.mult='r2l' 

sage: p213*p312 

[3, 2, 1] 

sage: Permutations.options.mult='l2r' 

""" 

if self.parent().options.mult == 'l2r': 

return self._left_to_right_multiply_on_right(rp) 

else: 

return self._left_to_right_multiply_on_left(rp) 

 

_mul_ = __mul__ # For ``prod()`` 

 

def __rmul__(self, lp): 

""" 

TESTS:: 

 

sage: p213 = Permutation([2,1,3]) 

sage: p312 = Permutation([3,1,2]) 

sage: Permutations.options.mult='l2r' 

sage: p213*p312 

[1, 3, 2] 

sage: Permutations.options.mult='r2l' 

sage: p213*p312 

[3, 2, 1] 

sage: Permutations.options.mult='l2r' 

""" 

if self.parent().options.mult == 'l2r': 

return self._left_to_right_multiply_on_left(lp) 

else: 

return self._left_to_right_multiply_on_right(lp) 

 

def left_action_product(self, lp): 

r""" 

Return the permutation obtained by composing ``self`` with 

``lp`` in such an order that ``lp`` is applied first and 

``self`` is applied afterwards. 

 

This is usually denoted by either ``self * lp`` or ``lp * self`` 

depending on the conventions used by the author. If the value 

of a permutation `p \in S_n` on an integer 

`i \in \{ 1, 2, \cdots, n \}` is denoted by `p(i)`, then this 

should be denoted by ``self * lp`` in order to have 

associativity (i.e., in order to have 

`(p \cdot q)(i) = p(q(i))` for all `p`, `q` and `i`). If, on 

the other hand, the value of a permutation `p \in S_n` on an 

integer `i \in \{ 1, 2, \cdots, n \}` is denoted by `i^p`, then 

this should be denoted by ``lp * self`` in order to have 

associativity (i.e., in order to have 

`i^{p \cdot q} = (i^p)^q` for all `p`, `q` and `i`). 

 

EXAMPLES:: 

 

sage: p = Permutation([2,1,3]) 

sage: q = Permutation([3,1,2]) 

sage: p.left_action_product(q) 

[3, 2, 1] 

sage: q.left_action_product(p) 

[1, 3, 2] 

""" 

return Permutations()(left_action_product(self._list, lp[:])) 

 

_left_to_right_multiply_on_left = left_action_product 

 

def right_action_product(self, rp): 

""" 

Return the permutation obtained by composing ``self`` with 

``rp`` in such an order that ``self`` is applied first and 

``rp`` is applied afterwards. 

 

This is usually denoted by either ``self * rp`` or ``rp * self`` 

depending on the conventions used by the author. If the value 

of a permutation `p \in S_n` on an integer 

`i \in \{ 1, 2, \cdots, n \}` is denoted by `p(i)`, then this 

should be denoted by ``rp * self`` in order to have 

associativity (i.e., in order to have 

`(p \cdot q)(i) = p(q(i))` for all `p`, `q` and `i`). If, on 

the other hand, the value of a permutation `p \in S_n` on an 

integer `i \in \{ 1, 2, \cdots, n \}` is denoted by `i^p`, then 

this should be denoted by ``self * rp`` in order to have 

associativity (i.e., in order to have 

`i^{p \cdot q} = (i^p)^q` for all `p`, `q` and `i`). 

 

EXAMPLES:: 

 

sage: p = Permutation([2,1,3]) 

sage: q = Permutation([3,1,2]) 

sage: p.right_action_product(q) 

[1, 3, 2] 

sage: q.right_action_product(p) 

[3, 2, 1] 

""" 

return Permutations()(right_action_product(self._list, rp[:])) 

 

_left_to_right_multiply_on_right = right_action_product 

 

def __call__(self, i): 

r""" 

Return the image of the integer `i` under ``self``. 

 

EXAMPLES:: 

 

sage: p = Permutation([2, 1, 4, 5, 3]) 

sage: p(1) 

2 

sage: p = Permutation(((1,2),(4,3,5))) 

sage: p(4) 

3 

sage: p(2) 

1 

sage: p = Permutation([5,2,1,6,3,7,4]) 

sage: list(map(p, range(1,8))) 

[5, 2, 1, 6, 3, 7, 4] 

 

TESTS:: 

 

sage: p = Permutation([5,2,1,6,3,7,4]) 

sage: p(-1) 

Traceback (most recent call last): 

... 

TypeError: i (= -1) must be between 1 and 7 

sage: p(10) 

Traceback (most recent call last): 

... 

TypeError: i (= 10) must be between 1 and 7 

""" 

if isinstance(i,(int,Integer)) and 1 <= i <= len(self): 

return self[i-1] 

else: 

raise TypeError("i (= %s) must be between %s and %s" % (i,1,len(self))) 

 

######## 

# Rank # 

######## 

 

def rank(self): 

r""" 

Return the rank of ``self`` in the lexicographic ordering on the 

symmetric group to which ``self`` belongs. 

 

EXAMPLES:: 

 

sage: Permutation([1,2,3]).rank() 

0 

sage: Permutation([1, 2, 4, 6, 3, 5]).rank() 

10 

sage: perms = Permutations(6).list() 

sage: [p.rank() for p in perms] == list(range(factorial(6))) 

True 

""" 

n = len(self) 

 

factoradic = self.to_lehmer_code() 

 

#Compute the index 

rank = 0 

for i in reversed(range(0, n)): 

rank += factoradic[n-1-i]*factorial(i) 

 

return rank 

 

############## 

# Inversions # 

############## 

 

def to_inversion_vector(self): 

r""" 

Return the inversion vector of ``self``. 

 

The inversion vector of a permutation `p \in S_n` is defined as 

the vector `(v_1, v_2, \ldots, v_n)`, where `v_i` is the 

number of elements larger than `i` that appear to the left 

of `i` in the permutation `p`. 

 

The algorithm is of complexity `O(n\log(n))` where `n` is the size of 

the given permutation. 

 

EXAMPLES:: 

 

sage: Permutation([5,9,1,8,2,6,4,7,3]).to_inversion_vector() 

[2, 3, 6, 4, 0, 2, 2, 1, 0] 

sage: Permutation([8,7,2,1,9,4,6,5,10,3]).to_inversion_vector() 

[3, 2, 7, 3, 4, 3, 1, 0, 0, 0] 

sage: Permutation([3,2,4,1,5]).to_inversion_vector() 

[3, 1, 0, 0, 0] 

 

TESTS:: 

 

sage: from sage.combinat.permutation import from_inversion_vector 

sage: all(from_inversion_vector(p.to_inversion_vector()) == p 

....: for n in range(6) for p in Permutations(n)) 

True 

 

sage: P = Permutations(1000) 

sage: sample = (P.random_element() for i in range(5)) 

sage: all(from_inversion_vector(p.to_inversion_vector()) == p 

....: for p in sample) 

True 

""" 

p = self._list 

l = len(p) 

# lightning fast if the length is less than 3 

# (is it really useful?) 

if l<4: 

if l==0: 

return [] 

if l==1: 

return [0] 

if l==2: 

return [p[0]-1,0] 

if l==3: 

if p[0]==1: 

return [0,p[1]-2,0] 

if p[0]==2: 

if p[1]==1: 

return [1,0,0] 

return [2,0,0] 

return [p[1],1,0] 

# choose the best one 

if l<411: 

return self._to_inversion_vector_small() 

else: 

return self._to_inversion_vector_divide_and_conquer() 

 

def _to_inversion_vector_orig(self): 

r""" 

Return the inversion vector of ``self``. 

 

The inversion vector of a permutation `p \in S_n` is defined as 

the vector `(v_1 , v_2 , \ldots , v_n)`, where `v_i` is the 

number of elements larger than `i` that appear to the left 

of `i` in the permutation `p`. 

 

(This implementation is probably not the most efficient one.) 

 

EXAMPLES:: 

 

sage: p = Permutation([5,9,1,8,2,6,4,7,3]) 

sage: p._to_inversion_vector_orig() 

[2, 3, 6, 4, 0, 2, 2, 1, 0] 

""" 

p = self._list 

iv = [0]*len(p) 

for i in range(len(p)): 

for pj in p: 

if pj>i+1: 

iv[i]+=1 

elif pj == i+1: 

break 

return iv 

 

def _to_inversion_vector_small(self): 

r""" 

Return the inversion vector of ``self``. 

 

The inversion vector of a permutation `p \in S_n` is defined as 

the vector `(v_1, v_2, \ldots, v_n)`, where `v_i` is the 

number of elements larger than `i` that appear to the left 

of `i` in the permutation `p`. 

 

(This implementation is the best choice for ``5 < size < 420`` 

approximately.) 

 

EXAMPLES:: 

 

sage: p = Permutation([5,9,1,8,2,6,4,7,3]) 

sage: p._to_inversion_vector_small() 

[2, 3, 6, 4, 0, 2, 2, 1, 0] 

""" 

p = self._list 

l = len(p)+1 

iv = [0]*l 

checked = [1]*l 

for pi in reversed(p): 

checked[pi] = 0 

iv[pi] = sum(checked[pi:]) 

return iv[1:] 

 

def _to_inversion_vector_divide_and_conquer(self): 

r""" 

Return the inversion vector of a permutation ``self``. 

 

The inversion vector of a permutation `p \in S_n` is defined as 

the vector `(v_1, v_2, \ldots, v_n)`, where `v_i` is the 

number of elements larger than `i` that appear to the left 

of `i` in the permutation `p`. 

 

(This implementation is the best choice for ``size > 410`` 

approximately.) 

 

EXAMPLES:: 

 

sage: p = Permutation([5,9,1,8,2,6,4,7,3]) 

sage: p._to_inversion_vector_divide_and_conquer() 

[2, 3, 6, 4, 0, 2, 2, 1, 0] 

""" 

# for big permutations, 

# we use a divide-and-conquer strategy 

# it's a merge sort, plus counting inversions 

def merge_and_countv(ivA_A, ivB_B): 

# iv* is the inversion vector of * 

(ivA, A) = ivA_A 

(ivB, B) = ivB_B 

C = [] 

i, j = 0, 0 

ivC = [] 

lA, lB = len(A), len(B) 

while(i < lA and j < lB): 

if B[j] < A[i]: 

C.append(B[j]) 

ivC.append(ivB[j] + lA - i) 

j += 1 

else: 

C.append(A[i]) 

ivC.append(ivA[i]) 

i += 1 

if i < lA: 

C.extend(A[i:]) 

ivC.extend(ivA[i:]) 

else: 

C.extend(B[j:]) 

ivC.extend(ivB[j:]) 

return ivC,C 

 

def base_case(L): 

s = sorted(L) 

d = dict((j,i) for (i,j) in enumerate(s)) 

iv = [0]*len(L) 

checked = [1]*len(L) 

for pi in reversed(L): 

dpi = d[pi] 

checked[dpi] = 0 

iv[dpi] = sum(checked[dpi:]) 

return iv,s 

 

def sort_and_countv(L): 

if len(L)<250: 

return base_case(L) 

l = len(L)//2 

return merge_and_countv( sort_and_countv(L[:l]), 

sort_and_countv(L[l:]) ) 

 

return sort_and_countv(self._list)[0] 

 

def inversions(self): 

r""" 

Return a list of the inversions of ``self``. 

 

An inversion of a permutation `p` is a pair `(i, j)` such that 

`i < j` and `p(i) > p(j)`. 

 

EXAMPLES:: 

 

sage: Permutation([3,2,4,1,5]).inversions() 

[(1, 2), (1, 4), (2, 4), (3, 4)] 

""" 

p = self[:] 

n = len(p) 

return [tuple([i+1,j+1]) for i in range(n-1) for j in range(i+1,n) 

if p[i]>p[j]] 

 

def show(self, representation="cycles", orientation="landscape", **args): 

r""" 

Display the permutation as a drawing. 

 

INPUT: 

 

- ``representation`` -- different kinds of drawings are available 

 

- ``"cycles"`` (default) -- the permutation is displayed as a 

collection of directed cycles 

 

- ``"braid"`` -- the permutation is displayed as segments linking 

each element `1, ..., n` to its image on a parallel line. 

 

When using this drawing, it is also possible to display the 

permutation horizontally (``orientation = "landscape"``, default 

option) or vertically (``orientation = "portrait"``). 

 

- ``"chord-diagram"`` -- the permutation is displayed as a directed 

graph, all of its vertices being located on a circle. 

 

All additional arguments are forwarded to the ``show`` subcalls. 

 

EXAMPLES:: 

 

sage: Permutations(20).random_element().show(representation = "cycles") 

sage: Permutations(20).random_element().show(representation = "chord-diagram") 

sage: Permutations(20).random_element().show(representation = "braid") 

sage: Permutations(20).random_element().show(representation = "braid", orientation='portrait') 

 

TESTS:: 

 

sage: Permutations(20).random_element().show(representation = "modern_art") 

Traceback (most recent call last): 

... 

ValueError: The value of 'representation' must be equal to 'cycles', 'chord-diagram' or 'braid' 

""" 

if representation == "cycles" or representation == "chord-diagram": 

d = DiGraph(loops = True) 

for i in range(len(self)): 

d.add_edge(i+1, self[i]) 

 

if representation == "cycles": 

d.show(**args) 

else: 

d.show(layout = "circular", **args) 

 

elif representation == "braid": 

from sage.plot.line import line 

from sage.plot.text import text 

 

if orientation == "landscape": 

r = lambda x,y : (x,y) 

elif orientation == "portrait": 

r = lambda x,y : (-y,x) 

else: 

raise ValueError("The value of 'orientation' must be either "+ 

"'landscape' or 'portrait'.") 

 

p = self[:] 

 

L = line([r(1,1)]) 

for i in range(len(p)): 

L += line([r(i,1.0), r(p[i]-1,0)]) 

L += text(str(i), r(i,1.05)) + text(str(i), r(p[i]-1,-.05)) 

 

return L.show(axes = False, **args) 

 

else: 

raise ValueError("The value of 'representation' must be equal to "+ 

"'cycles', 'chord-diagram' or 'braid'") 

 

 

def number_of_inversions(self): 

r""" 

Return the number of inversions in ``self``. 

 

An inversion of a permutation is a pair of elements `(i, j)` 

with `i < j` and `p(i) > p(j)`. 

 

REFERENCES: 

 

- http://mathworld.wolfram.com/PermutationInversion.html 

 

EXAMPLES:: 

 

sage: Permutation([3, 2, 4, 1, 5]).number_of_inversions() 

4 

sage: Permutation([1, 2, 6, 4, 7, 3, 5]).number_of_inversions() 

6 

""" 

return sum(self.to_inversion_vector()) 

 

def noninversions(self, k): 

r""" 

Return the list of all ``k``-noninversions in ``self``. 

 

If `k` is an integer and `p \in S_n` is a permutation, then 

a `k`-noninversion in `p` is defined as a strictly increasing 

sequence `(i_1, i_2, \ldots, i_k)` of elements of 

`\{ 1, 2, \ldots, n \}` satisfying 

`p(i_1) < p(i_2) < \cdots < p(i_k)`. (In other words, a 

`k`-noninversion in `p` can be regarded as a `k`-element 

subset of `\{ 1, 2, \ldots, n \}` on which `p` restricts 

to an increasing map.) 

 

EXAMPLES:: 

 

sage: p = Permutation([3, 2, 4, 1, 5]) 

sage: p.noninversions(1) 

[[3], [2], [4], [1], [5]] 

sage: p.noninversions(2) 

[[3, 4], [3, 5], [2, 4], [2, 5], [4, 5], [1, 5]] 

sage: p.noninversions(3) 

[[3, 4, 5], [2, 4, 5]] 

sage: p.noninversions(4) 

[] 

sage: p.noninversions(5) 

[] 

 

TESTS:: 

 

sage: q = Permutation([]) 

sage: q.noninversions(1) 

[] 

""" 

if k > len(self): 

return [] 

return [list(pos) for pos in itertools.combinations(self, k) if all( pos[i] < pos[i+1] for i in range(k-1) )] 

 

def number_of_noninversions(self, k): 

r""" 

Return the number of ``k``-noninversions in ``self``. 

 

If `k` is an integer and `p \in S_n` is a permutation, then 

a `k`-noninversion in `p` is defined as a strictly increasing 

sequence `(i_1, i_2, \ldots, i_k)` of elements of 

`\{ 1, 2, \ldots, n \}` satisfying 

`p(i_1) < p(i_2) < \cdots < p(i_k)`. (In other words, a 

`k`-noninversion in `p` can be regarded as a `k`-element 

subset of `\{ 1, 2, \ldots, n \}` on which `p` restricts 

to an increasing map.) 

 

The number of `k`-noninversions in `p` has been denoted by 

`\mathrm{noninv}_k(p)` in [RSW2011]_, where conjectures 

and results regarding this number have been stated. 

 

REFERENCES: 

 

.. [RSW2011] Victor Reiner, Franco Saliola, Volkmar Welker. 

*Spectra of Symmetrized Shuffling Operators*. 

:arXiv:`1102.2460v2`. 

 

EXAMPLES:: 

 

sage: p = Permutation([3, 2, 4, 1, 5]) 

sage: p.number_of_noninversions(1) 

5 

sage: p.number_of_noninversions(2) 

6 

sage: p.number_of_noninversions(3) 

2 

sage: p.number_of_noninversions(4) 

0 

sage: p.number_of_noninversions(5) 

0 

 

The number of `2`-noninversions of a permutation `p \in S_n` 

is `\binom{n}{2}` minus its number of inversions:: 

 

sage: b = binomial(5, 2) 

sage: all( x.number_of_noninversions(2) == b - x.number_of_inversions() 

....: for x in Permutations(5) ) 

True 

 

We also check some corner cases:: 

 

sage: all( x.number_of_noninversions(1) == 5 for x in Permutations(5) ) 

True 

sage: all( x.number_of_noninversions(0) == 1 for x in Permutations(5) ) 

True 

sage: Permutation([]).number_of_noninversions(1) 

0 

sage: Permutation([]).number_of_noninversions(0) 

1 

sage: Permutation([2, 1]).number_of_noninversions(3) 

0 

""" 

if k > len(self): 

return 0 

return sum(1 for pos in itertools.combinations(self, k) 

if all(pos[i] < pos[i + 1] for i in range(k - 1))) 

 

def length(self): 

r""" 

Return the Coxeter length of ``self``. 

 

The length of a permutation `p` is given by the number of inversions 

of `p`. 

 

EXAMPLES:: 

 

sage: Permutation([5, 1, 3, 4, 2]).length() 

6 

""" 

return self.number_of_inversions() 

 

def absolute_length(self): 

""" 

Return the absolute length of ``self`` 

 

The absolute length is the length of the shortest expression 

of the element as a product of reflections. 

 

For permutations in the symmetric groups, the absolute 

length is the size minus the number of its disjoint 

cycles. 

 

EXAMPLES:: 

 

sage: Permutation([4,2,3,1]).absolute_length() 

1 

""" 

return self.size() - len(self.cycle_type()) 

 

@combinatorial_map(order=2,name='inverse') 

def inverse(self): 

r""" 

Return the inverse of ``self``. 

 

EXAMPLES:: 

 

sage: Permutation([3,8,5,10,9,4,6,1,7,2]).inverse() 

[8, 10, 1, 6, 3, 7, 9, 2, 5, 4] 

sage: Permutation([2, 4, 1, 5, 3]).inverse() 

[3, 1, 5, 2, 4] 

sage: ~Permutation([2, 4, 1, 5, 3]) 

[3, 1, 5, 2, 4] 

""" 

w = list(range(len(self))) 

for i,j in enumerate(self): 

w[j-1] = i+1 

return Permutations()(w) 

 

__invert__ = inverse 

 

def _icondition(self, i): 

""" 

Return a string which shows the relative positions of `i-1,i,i+1` in 

``self``, along with the actual positions of these three letters in 

``self``. The string represents the letters `i-1,i,i+1` by `1,2,3`, 

respectively. 

 

.. NOTE:: 

 

An imove (that is, an iswitch or an ishift) can only be applied 

when the relative positions of `i-1,i,i+1` are one of '213', 

'132', '231', or '312'. ``None`` is returned in the other cases 

to signal that an imove cannot be applied. 

 

EXAMPLES:: 

 

sage: Permutation([2,1,3])._icondition(2) 

('213', 1, 0, 2) 

sage: Permutation([1,3,2])._icondition(2) 

('132', 0, 2, 1) 

sage: Permutation([2,3,1])._icondition(2) 

('231', 2, 0, 1) 

sage: Permutation([3,1,2])._icondition(2) 

('312', 1, 2, 0) 

sage: Permutation([1,2,3])._icondition(2) 

(None, 0, 1, 2) 

sage: Permutation([1,3,2,4])._icondition(3) 

('213', 2, 1, 3) 

sage: Permutation([2,1,3])._icondition(3) 

Traceback (most recent call last): 

... 

ValueError: i (= 3) must be between 2 and n-1 

 

.. SEEALSO:: 

 

:meth:`ishift`, :meth:`iswitch` 

""" 

if i not in range(2, len(self)): 

raise ValueError("i (= %s) must be between 2 and n-1"%i) 

pos_i = self.index(i) 

pos_ip1 = self.index(i+1) 

pos_im1 = self.index(i-1) 

 

if pos_i < pos_im1 and pos_im1 < pos_ip1: 

state = '213' 

elif pos_im1 < pos_ip1 and pos_ip1 < pos_i: 

state = '132' 

elif pos_i < pos_ip1 and pos_ip1 < pos_im1: 

state = '231' 

elif pos_ip1 < pos_im1 and pos_im1 < pos_i: 

state = '312' 

else: 

state = None 

 

return (state, pos_im1, pos_i, pos_ip1) 

 

def ishift(self, i): 

""" 

Return the ``i``-shift of ``self``. If an ``i``-shift of ``self`` 

can't be performed, then ``self`` is returned. 

 

An `i`-shift can be applied when `i` is not inbetween `i-1` and 

`i+1`. The `i`-shift moves `i` to the other side, and leaves the 

relative positions of `i-1` and `i+1` in place. All other entries 

of the permutations are also left in place. 

 

EXAMPLES: 

 

Here, `2` is to the left of both `1` and `3`. A `2`-shift 

can be applied which moves the `2` to the right and leaves `1` and 

`3` in their same relative order:: 

 

sage: Permutation([2,1,3]).ishift(2) 

[1, 3, 2] 

 

All entries other than `i`, `i-1` and `i+1` are unchanged:: 

 

sage: Permutation([2,4,1,3]).ishift(2) 

[1, 4, 3, 2] 

 

Since `2` is between `1` and `3` in ``[1,2,3]``, a `2`-shift cannot 

be applied to ``[1,2,3]`` :: 

 

sage: Permutation([1,2,3]).ishift(2) 

[1, 2, 3] 

""" 

state = self._icondition(i) 

if state[0] is None: 

return self 

 

state, pos_im1, pos_i, pos_ip1 = state 

l = list(self) 

 

if state == '213': #goes to 132 

l[pos_i] = i-1 

l[pos_im1] = i+1 

l[pos_ip1] = i 

elif state == '132': #goes to 213 

l[pos_im1] = i 

l[pos_ip1] = i-1 

l[pos_i] = i+1 

elif state == '231': #goes to 312 

l[pos_i] = i+1 

l[pos_ip1] = i-1 

l[pos_im1] = i 

elif state == '312': #goes to 231 

l[pos_ip1] = i 

l[pos_im1] = i+1 

l[pos_i] = i-1 

else: 

# This branch should never occur, no matter what the user does. 

raise ValueError("invalid state") 

 

return Permutations()(l) 

 

def iswitch(self, i): 

""" 

Return the ``i``-switch of ``self``. If an ``i``-switch of ``self`` 

can't be performed, then ``self`` is returned. 

 

An `i`-switch can be applied when the subsequence of ``self`` formed 

by the entries `i-1`, `i` and `i+1` is neither increasing nor 

decreasing. In this case, this subsequence is reversed (i. e., its 

leftmost element and its rightmost element switch places), while all 

other letters of ``self`` are kept in place. 

 

EXAMPLES: 

 

Here, `2` is to the left of both `1` and `3`. A `2`-switch can be 

applied which moves the `2` to the right and switches the relative 

order between `1` and `3`:: 

 

sage: Permutation([2,1,3]).iswitch(2) 

[3, 1, 2] 

 

All entries other than `i-1`, `i` and `i+1` are unchanged:: 

 

sage: Permutation([2,4,1,3]).iswitch(2) 

[3, 4, 1, 2] 

 

Since `2` is between `1` and `3` in ``[1,2,3]``, a `2`-switch 

cannot be applied to ``[1,2,3]`` :: 

 

sage: Permutation([1,2,3]).iswitch(2) 

[1, 2, 3] 

""" 

if i not in range(2, len(self)): 

raise ValueError("i (= %s) must between 2 and n-1"%i) 

 

state = self._icondition(i) 

if state[0] is None: 

return self 

 

state, pos_im1, pos_i, pos_ip1 = state 

l = list(self) 

 

if state == '213': #goes to 312 

l[pos_i] = i+1 

l[pos_ip1] = i 

elif state == '132': #goes to 231 

l[pos_im1] = i 

l[pos_i] = i-1 

elif state == '231': #goes to 132 

l[pos_i] = i-1 

l[pos_im1] = i 

elif state == '312': #goes to 213 

l[pos_ip1] = i 

l[pos_i] = i+1 

else: 

# This branch should never occur, no matter what the user does. 

raise ValueError("invalid state") 

 

return Permutations()(l) 

 

def runs(self, as_tuple=False): 

r""" 

Return a list of the runs in the nonempty permutation 

``self``. 

 

A run in a permutation is defined to be a maximal (with 

respect to inclusion) nonempty increasing substring (i. e., 

contiguous subsequence). For instance, the runs in the 

permutation ``[6,1,7,3,4,5,2]`` are ``[6]``, ``[1,7]``, 

``[3,4,5]`` and ``[2]``. 

 

Runs in an empty permutation are not defined. 

 

INPUT: 

 

- ``as_tuple`` -- boolean (default: ``False``) choice of 

output format 

 

OUTPUT: 

 

a list of lists or a tuple of tuples 

 

REFERENCES: 

 

- http://mathworld.wolfram.com/PermutationRun.html 

 

EXAMPLES:: 

 

sage: Permutation([1,2,3,4]).runs() 

[[1, 2, 3, 4]] 

sage: Permutation([4,3,2,1]).runs() 

[[4], [3], [2], [1]] 

sage: Permutation([2,4,1,3]).runs() 

[[2, 4], [1, 3]] 

sage: Permutation([1]).runs() 

[[1]] 

 

The example from above:: 

 

sage: Permutation([6,1,7,3,4,5,2]).runs() 

[[6], [1, 7], [3, 4, 5], [2]] 

sage: Permutation([6,1,7,3,4,5,2]).runs(as_tuple=True) 

((6,), (1, 7), (3, 4, 5), (2,)) 

 

The number of runs in a nonempty permutation equals its 

number of descents plus 1:: 

 

sage: all( len(p.runs()) == p.number_of_descents() + 1 

....: for p in Permutations(6) ) 

True 

""" 

p = self[:] 

runs = [] 

current_value = p[0] 

current_run = [p[0]] 

for i in p[1:]: 

if i < current_value: 

runs.append(current_run) 

current_run = [i] 

else: 

current_run.append(i) 

 

current_value = i 

runs.append(current_run) 

 

if as_tuple: 

return tuple(tuple(r) for r in runs) 

 

return runs 

 

def decreasing_runs(self, as_tuple=False): 

""" 

Decreasing runs of the permutation. 

 

INPUT: 

 

- ``as_tuple`` -- boolean (default: ``False``) choice of output 

format 

 

OUTPUT: 

 

a list of lists or a tuple of tuples 

 

.. SEEALSO:: 

 

:meth:`runs` 

 

EXAMPLES:: 

 

sage: s = Permutation([2,8,3,9,6,4,5,1,7]) 

sage: s.decreasing_runs() 

[[2], [8, 3], [9, 6, 4], [5, 1], [7]] 

sage: s.decreasing_runs(as_tuple=True) 

((2,), (8, 3), (9, 6, 4), (5, 1), (7,)) 

""" 

n = len(self) 

s_bar = self.complement() 

if as_tuple: 

return tuple(tuple(n + 1 - i for i in r) for r in s_bar.runs()) 

return [[n + 1 - i for i in r] for r in s_bar.runs()] 

 

def longest_increasing_subsequence_length(self): 

r""" 

Return the length of the longest increasing subsequences of ``self``. 

 

EXAMPLES:: 

 

sage: Permutation([2,3,1,4]).longest_increasing_subsequence_length() 

3 

sage: all(i.longest_increasing_subsequence_length() == len(RSK(i)[0][0]) for i in Permutations(5)) 

True 

sage: Permutation([]).longest_increasing_subsequence_length() 

0 

""" 

r=[] 

for x in self: 

if max(r+[0]) > x: 

y = min(z for z in r if z > x) 

r[r.index(y)] = x 

else: 

r.append(x) 

return len(r) 

 

def longest_increasing_subsequences(self): 

r""" 

Return the list of the longest increasing subsequences of ``self``. 

 

.. note:: 

 

The algorithm is not optimal. 

 

EXAMPLES:: 

 

sage: Permutation([2,3,4,1]).longest_increasing_subsequences() 

[[2, 3, 4]] 

sage: Permutation([5, 7, 1, 2, 6, 4, 3]).longest_increasing_subsequences() 

[[1, 2, 6], [1, 2, 4], [1, 2, 3]] 

""" 

patt = list(range(1,self.longest_increasing_subsequence_length()+1)) 

return [[self[i] for i in m] for m in self.pattern_positions(patt)] 

 

def cycle_type(self): 

r""" 

Return a partition of ``len(self)`` corresponding to the cycle 

type of ``self``. 

This is a non-increasing sequence of the cycle lengths of ``self``. 

 

EXAMPLES:: 

 

sage: Permutation([3,1,2,4]).cycle_type() 

[3, 1] 

""" 

cycle_type = [len(c) for c in self.to_cycles()] 

cycle_type.sort(reverse=True) 

from sage.combinat.partition import Partition 

return Partition(cycle_type) 

 

@combinatorial_map(name='forget cycles') 

def forget_cycles(self): 

r""" 

Return the image of ``self`` under the map which forgets cycles. 

 

Consider a permutation `\sigma` written in standard cyclic form: 

 

.. MATH:: 

 

\sigma = (a_{1,1}, \ldots, a_{1,k_1}) (a_{2,1}, \ldots, a_{2,k_2}) 

\cdots (a_{m,1}, \ldots, a_{m,k_m}), 

 

where `a_{1,1} < a_{2,1} < \cdots < a_{m,1}` and `a_{j,1} < a_{j,i}` 

for all `1 \leq j \leq m` and `2 \leq i \leq k_j` where we include 

cycles of length 1 as well. The image of the forget cycle map `\phi` 

is given by 

 

.. MATH:: 

 

\phi(\sigma) = [a_{1,1}, \ldots, a_{1,k_1}, a_{2,1} \ldots, 

a_{2,k_2}, \ldots, a_{m,1}, \ldots, a_{m,k_m}], 

 

considered as a permutation in 1-line notation. 

 

EXAMPLES:: 

 

sage: P = Permutations(5) 

sage: x = P([1, 5, 3, 4, 2]) 

sage: x.forget_cycles() 

[1, 2, 5, 3, 4] 

 

We select all permutations with a cycle composition of `[2, 3, 1]` 

in `S_6`:: 

 

sage: P = Permutations(6) 

sage: l = [p for p in P if [len(t) for t in p.to_cycles()] == [1,3,2]] 

 

Next we apply `\phi` and then take the inverse, and then view the 

results as a poset under the Bruhat order:: 

 

sage: l = [p.forget_cycles().inverse() for p in l] 

sage: B = Poset([l, lambda x,y: x.bruhat_lequal(y)]) 

sage: R.<q> = QQ[] 

sage: sum(q^B.rank_function()(x) for x in B) 

q^5 + 2*q^4 + 3*q^3 + 3*q^2 + 2*q + 1 

 

We check the statement in [CC13]_ that the posets 

`C_{[1,3,1,1]}` and `C_{[1,3,2]}` are isomorphic:: 

 

sage: l2 = [p for p in P if [len(t) for t in p.to_cycles()] == [1,3,1,1]] 

sage: l2 = [p.forget_cycles().inverse() for p in l2] 

sage: B2 = Poset([l2, lambda x,y: x.bruhat_lequal(y)]) 

sage: B.is_isomorphic(B2) 

True 

 

REFERENCES: 

 

.. [CC13] Mahir Bilen Can and Yonah Cherniavsky. 

*Omitting parentheses from the cyclic notation*. (2013). 

:arxiv:`1308.0936v2`. 

""" 

ret = [] 

for t in self.to_cycles(): 

ret += list(t) 

return Permutations()(ret) 

 

@combinatorial_map(name='foata_bijection') 

def foata_bijection(self): 

r""" 

Return the image of the permutation ``self`` under the Foata 

bijection `\phi`. 

 

The bijection shows that `\mathrm{maj}` and `\mathrm{inv}` are 

equidistributed: if `\phi(P) = Q`, then `\mathrm{maj}(P) = 

\mathrm{inv}(Q)`. 

 

The Foata bijection `\phi` is a bijection on the set of words with 

no two equal letters. It can be defined by induction on the size 

of the word: Given a word `w_1 w_2 \cdots w_n`, start with 

`\phi(w_1) = w_1`. At the `i`-th step, if 

`\phi(w_1 w_2 \cdots w_i) = v_1 v_2 \cdots v_i`, we define 

`\phi(w_1 w_2 \cdots w_i w_{i+1})` by placing `w_{i+1}` on the end of 

the word `v_1 v_2 \cdots v_i` and breaking the word up into blocks 

as follows. If `w_{i+1} > v_i`, place a vertical line to the right 

of each `v_k` for which `w_{i+1} > v_k`. Otherwise, if 

`w_{i+1} < v_i`, place a vertical line to the right of each `v_k` 

for which `w_{i+1} < v_k`. In either case, place a vertical line at 

the start of the word as well. Now, within each block between 

vertical lines, cyclically shift the entries one place to the 

right. 

 

For instance, to compute `\phi([1,4,2,5,3])`, the sequence of 

words is 

 

* `1`, 

* `|1|4 \to 14`, 

* `|14|2 \to 412`, 

* `|4|1|2|5 \to 4125`, 

* `|4|125|3 \to 45123`. 

 

So `\phi([1,4,2,5,3]) = [4,5,1,2,3]`. 

 

See section 2 of [FoSc78]_. 

 

REFERENCES: 

 

.. [FoSc78] Dominique Foata, Marcel-Paul Schuetzenberger. 

*Major Index and Inversion Number of Permutations*. 

Mathematische Nachrichten, volume 83, Issue 1, pages 143-159, 1978. 

http://igm.univ-mlv.fr/~berstel/Mps/Travaux/A/1978-3MajorIndexMathNachr.pdf 

 

EXAMPLES:: 

 

sage: Permutation([1,2,4,3]).foata_bijection() 

[4, 1, 2, 3] 

sage: Permutation([2,5,1,3,4]).foata_bijection() 

[2, 1, 3, 5, 4] 

 

sage: P = Permutation([2,5,1,3,4]) 

sage: P.major_index() == P.foata_bijection().number_of_inversions() 

True 

 

sage: all( P.major_index() == P.foata_bijection().number_of_inversions() 

....: for P in Permutations(4) ) 

True 

 

The example from [FoSc78]_:: 

 

sage: Permutation([7,4,9,2,6,1,5,8,3]).foata_bijection() 

[4, 7, 2, 6, 1, 9, 5, 8, 3] 

 

Border cases:: 

 

sage: Permutation([]).foata_bijection() 

[] 

sage: Permutation([1]).foata_bijection() 

[1] 

""" 

L = list(self) 

M = [] 

for e in L: 

M.append(e) 

k = len(M) 

if k <= 1: 

continue 

 

a = M[-2] 

M_prime = [0]*k 

if a > e: 

index_list = [-1] + [i for i in range(k - 1) if M[i] > e] 

else: 

index_list = [-1] + [i for i in range(k - 1) if M[i] < e] 

 

for j in range(1, len(index_list)): 

start = index_list[j-1] + 1 

end = index_list[j] 

M_prime[start] = M[end] 

for x in range(start + 1, end + 1): 

M_prime[x] = M[x-1] 

M_prime[k-1] = e 

M = M_prime 

return Permutations()(M) 

 

def destandardize(self, weight, ordered_alphabet = None): 

r""" 

Return destandardization of ``self`` with respect to ``weight`` and ``ordered_alphabet``. 

 

INPUT: 

 

- ``weight`` -- list or tuple of nonnegative integers that sum to `n` if ``self`` 

is a permutation in `S_n`. 

 

- ``ordered_alphabet`` -- (default: None) a list or tuple specifying the ordered alphabet the 

destandardized word is over 

 

OUTPUT: word over the ``ordered_alphabet`` which standardizes to ``self`` 

 

Let `weight = (w_1,w_2,\ldots,w_\ell)`. Then this methods looks for an increasing 

sequence of `1,2,\ldots, w_1` and labels all letters in it by 1, then an increasing 

sequence of `w_1+1,w_1+2,\ldots,w_1+w_2` and labels all these letters by 2, etc.. 

If an increasing sequence for the specified ``weight`` does not exist, an error is 

returned. The output is a word ``w`` over the specified ordered alphabet with 

evaluation ``weight`` such that ``w.standard_permutation()`` is ``self``. 

 

EXAMPLES:: 

 

sage: p = Permutation([1,2,5,3,6,4]) 

sage: p.destandardize([3,1,2]) 

word: 113132 

sage: p = Permutation([2,1,3]) 

sage: p.destandardize([2,1]) 

Traceback (most recent call last): 

... 

ValueError: Standardization with weight [2, 1] is not possible! 

 

TESTS:: 

 

sage: p = Permutation([4,1,2,3,5,6]) 

sage: p.destandardize([2,1,3], ordered_alphabet = [1,'a',3]) 

word: 311a33 

sage: p.destandardize([2,1,3], ordered_alphabet = [1,'a']) 

Traceback (most recent call last): 

... 

ValueError: Not enough letters in the alphabet are specified compared to the weight 

""" 

ides = self.idescents() 

partial = [0] 

for a in weight: 

partial.append(partial[-1]+a) 

if not set(ides).issubset(set(partial)): 

raise ValueError("Standardization with weight {} is not possible!".format(weight)) 

if ordered_alphabet is None: 

ordered_alphabet = list(range(1,len(weight)+1)) 

else: 

if len(weight) > len(ordered_alphabet): 

raise ValueError("Not enough letters in the alphabet are specified compared to the weight") 

q = self.inverse() 

s = [0]*len(self) 

for i in range(len(partial)-1): 

for j in range(partial[i],partial[i+1]): 

s[q[j]-1] = ordered_alphabet[i] 

from sage.combinat.words.word import Word 

return Word(s) 

 

def to_lehmer_code(self): 

r""" 

Return the Lehmer code of the permutation ``self``. 

 

The Lehmer code of a permutation `p` is defined as the 

list `[c[1],c[2],...,c[n]]`, where `c[i]` is the number of 

`j>i` such that `p(j)<p(i)`. 

 

EXAMPLES:: 

 

sage: p = Permutation([2,1,3]) 

sage: p.to_lehmer_code() 

[1, 0, 0] 

sage: q = Permutation([3,1,2]) 

sage: q.to_lehmer_code() 

[2, 0, 0] 

 

sage: Permutation([1]).to_lehmer_code() 

[0] 

sage: Permutation([]).to_lehmer_code() 

[] 

 

TESTS:: 

 

sage: from sage.combinat.permutation import from_lehmer_code 

sage: all(from_lehmer_code(p.to_lehmer_code()) == p 

....: for n in range(6) for p in Permutations(n)) 

True 

 

sage: P = Permutations(1000) 

sage: sample = (P.random_element() for i in range(5)) 

sage: all(from_lehmer_code(p.to_lehmer_code()) == p 

....: for p in sample) 

True 

 

""" 

l = len(self._list) 

# choose the best implementations 

if l<577: 

return self._to_lehmer_code_small() 

else: 

return self.inverse().to_inversion_vector() 

 

def _to_lehmer_code_small(self): 

r""" 

Return the Lehmer code of the permutation ``self``. 

 

The Lehmer code of a permutation `p` is defined as the 

list `(c_1, c_2, \ldots, c_n)`, where `c_i` is the number 

of `j > i` such that `p(j) < p(i)`. 

 

(best choice for `size<577` approximately) 

 

EXAMPLES:: 

 

sage: p = Permutation([7, 6, 10, 2, 3, 4, 8, 1, 9, 5]) 

sage: p._to_lehmer_code_small() 

[6, 5, 7, 1, 1, 1, 2, 0, 1, 0] 

""" 

p = self._list 

l = len(p) 

lehmer = [] 

checked = [1]*l 

for pi in p: 

checked[pi-1] = 0 

lehmer.append(sum(checked[:pi])) 

return lehmer 

 

def to_lehmer_cocode(self): 

r""" 

Return the Lehmer cocode of the permutation ``self``. 

 

The Lehmer cocode of a permutation `p` is defined as the 

list `(c_1, c_2, \ldots, c_n)`, where `c_i` is the number 

of `j < i` such that `p(j) > p(i)`. 

 

EXAMPLES:: 

 

sage: p = Permutation([2,1,3]) 

sage: p.to_lehmer_cocode() 

[0, 1, 0] 

sage: q = Permutation([3,1,2]) 

sage: q.to_lehmer_cocode() 

[0, 1, 1] 

""" 

p = self[:] 

n = len(p) 

cocode = [0] * n 

for i in range(1, n): 

for j in range(0, i): 

if p[j] > p[i]: 

cocode[i] += 1 

return cocode 

 

 

 

################# 

# Reduced Words # 

################# 

 

def reduced_word(self): 

r""" 

Return a reduced word of the permutation ``self``. 

 

See :meth:`reduced_words` for the definition of reduced words and 

a way to compute them all. 

 

.. WARNING:: 

 

This does not respect the multiplication convention. 

 

EXAMPLES:: 

 

sage: Permutation([3,5,4,6,2,1]).reduced_word() 

[2, 1, 4, 3, 2, 4, 3, 5, 4, 5] 

 

Permutation([1]).reduced_word_lexmin() 

[] 

Permutation([]).reduced_word_lexmin() 

[] 

""" 

code = self.to_lehmer_code() 

reduced_word = [] 

for piece in [ [ i + code[i] - j for j in range(code[i])] for i in range(len(code)) ]: 

reduced_word += piece 

 

return reduced_word 

 

def reduced_words_iterator(self): 

r""" 

Return an iterator for the reduced words of ``self``. 

 

EXAMPLES:: 

 

sage: next(Permutation([5,2,3,4,1]).reduced_words_iterator()) 

[1, 2, 3, 4, 3, 2, 1] 

 

""" 

def aux(p): 

is_identity = True 

for d in range(len(p)-1): 

e = d+1 

if p[d] > p[e]: 

is_identity = False 

p[d], p[e] = p[e], p[d] 

 

for x in aux(p): 

x.append(e) 

yield x 

 

p[d], p[e] = p[e], p[d] 

if is_identity: 

yield [] 

 

return aux(self[:]) 

 

def reduced_words(self): 

r""" 

Return a list of the reduced words of ``self``. 

 

The notion of a reduced word is based on the well-known fact 

that every permutation can be written as a product of adjacent 

transpositions. In more detail: If `n` is a nonnegative integer, 

we can define the transpositions `s_i = (i, i+1) \in S_n` 

for all `i \in \{ 1, 2, \ldots, n-1 \}`, and every `p \in S_n` 

can then be written as a product `s_{i_1} s_{i_2} \cdots s_{i_k}` 

for some sequence `(i_1, i_2, \ldots, i_k)` of elements of 

`\{ 1, 2, \ldots, n-1 \}` (here `\{ 1, 2, \ldots, n-1 \}` denotes 

the empty set when `n \leq 1`). Fixing a `p`, the sequences 

`(i_1, i_2, \ldots, i_k)` of smallest length satisfying 

`p = s_{i_1} s_{i_2} \cdots s_{i_k}` are called the reduced words 

of `p`. (Their length is the Coxeter length of `p`, and can be 

computed using :meth:`length`.) 

 

Note that the product of permutations is defined here in such 

a way that `(pq)(i) = p(q(i))` for all permutations `p` and `q` 

and each `i \in \{ 1, 2, \ldots, n \}` (this is the same 

convention as in :meth:`left_action_product`, but not the 

default semantics of the `*` operator on permutations in Sage). 

Thus, for instance, `s_2 s_1` is the permutation obtained by 

first transposing `1` with `2` and then transposing `2` with `3`. 

 

.. SEEALSO:: 

 

:meth:`reduced_word`, :meth:`reduced_word_lexmin` 

 

EXAMPLES:: 

 

sage: Permutation([2,1,3]).reduced_words() 

[[1]] 

sage: Permutation([3,1,2]).reduced_words() 

[[2, 1]] 

sage: Permutation([3,2,1]).reduced_words() 

[[1, 2, 1], [2, 1, 2]] 

sage: Permutation([3,2,4,1]).reduced_words() 

[[1, 2, 3, 1], [1, 2, 1, 3], [2, 1, 2, 3]] 

 

Permutation([1]).reduced_words() 

[[]] 

Permutation([]).reduced_words() 

[[]] 

""" 

return list(self.reduced_words_iterator()) 

 

def reduced_word_lexmin(self): 

r""" 

Return a lexicographically minimal reduced word of the permutation 

``self``. 

 

See :meth:`reduced_words` for the definition of reduced words and 

a way to compute them all. 

 

EXAMPLES:: 

 

sage: Permutation([3,4,2,1]).reduced_word_lexmin() 

[1, 2, 1, 3, 2] 

 

Permutation([1]).reduced_word_lexmin() 

[] 

Permutation([]).reduced_word_lexmin() 

[] 

""" 

cocode = self.inverse().to_lehmer_cocode() 

 

rw = [] 

for i in range(len(cocode)): 

piece = [j+1 for j in range(i-cocode[i],i)] 

piece.reverse() 

rw += piece 

 

return rw 

 

 

################ 

# Fixed Points # 

################ 

 

def fixed_points(self): 

r""" 

Return a list of the fixed points of ``self``. 

 

EXAMPLES:: 

 

sage: Permutation([1,3,2,4]).fixed_points() 

[1, 4] 

sage: Permutation([1,2,3,4]).fixed_points() 

[1, 2, 3, 4] 

""" 

fixed_points = [] 

for i in range(len(self)): 

if i+1 == self[i]: 

fixed_points.append(i+1) 

 

return fixed_points 

 

def number_of_fixed_points(self): 

r""" 

Return the number of fixed points of ``self``. 

 

EXAMPLES:: 

 

sage: Permutation([1,3,2,4]).number_of_fixed_points() 

2 

sage: Permutation([1,2,3,4]).number_of_fixed_points() 

4 

""" 

 

return len(self.fixed_points()) 

 

 

############ 

# Recoils # 

############ 

def recoils(self): 

r""" 

Return the list of the positions of the recoils of ``self``. 

 

A recoil of a permutation `p` is an integer `i` such that `i+1` 

appears to the left of `i` in `p`. 

Here, the positions are being counted starting at `0`. 

(Note that it is the positions, not the recoils themselves, which 

are being listed.) 

 

EXAMPLES:: 

 

sage: Permutation([1,4,3,2]).recoils() 

[2, 3] 

sage: Permutation([]).recoils() 

[] 

""" 

p = self 

recoils = [] 

for i in range(len(p)): 

if p[i] != len(self) and self.index(p[i]+1) < i: 

recoils.append(i) 

 

return recoils 

 

def number_of_recoils(self): 

r""" 

Return the number of recoils of the permutation ``self``. 

 

EXAMPLES:: 

 

sage: Permutation([1,4,3,2]).number_of_recoils() 

2 

""" 

return len(self.recoils()) 

 

def recoils_composition(self): 

r""" 

Return the recoils composition of ``self``. 

 

The recoils composition of a permutation `p \in S_n` is the 

composition of `n` whose descent set is the set of the recoils 

of `p` (not their positions). In other words, this is the 

descents composition of `p^{-1}`. 

 

EXAMPLES:: 

 

sage: Permutation([1,3,2,4]).recoils_composition() 

[2, 2] 

sage: Permutation([]).recoils_composition() 

[] 

""" 

return self.inverse().descents_composition() 

 

 

############ 

# Descents # 

############ 

 

def descents(self, final_descent=False, side='right', positive=False, 

from_zero=False, index_set=None): 

r""" 

Return the list of the descents of ``self``. 

 

A descent of a permutation `p` is an integer `i` such that 

`p(i) > p(i+1)`. 

 

.. WARNING:: 

 

By default, the descents are returned as elements in the 

index set, i.e., starting at `1`. If you want them to 

start at `0`, set the keyword ``from_zero`` to ``True``. 

 

INPUT: 

 

- ``final_descent`` -- boolean (default ``False``); 

if ``True``, the last position of a non-empty 

permutation is also considered as a descent 

 

- ``side`` -- ``'right'`` (default) or ``'left'``; 

if ``'left'``, return the descents of the inverse permutation 

 

- ``positive`` -- boolean (default ``False``); 

if ``True``, return the positions that are not descents 

 

- ``from_zero`` -- boolean (default ``False``); 

if ``True``, return the positions starting from `0` 

 

- ``index_set`` -- list (default: ``[1, ..., n-1]`` where ``self`` 

is a permutation of ``n``); the index set to check for descents 

 

EXAMPLES:: 

 

sage: Permutation([3,1,2]).descents() 

[1] 

sage: Permutation([1,4,3,2]).descents() 

[2, 3] 

sage: Permutation([1,4,3,2]).descents(final_descent=True) 

[2, 3, 4] 

sage: Permutation([1,4,3,2]).descents(index_set=[1,2]) 

[2] 

sage: Permutation([1,4,3,2]).descents(from_zero=True) 

[1, 2] 

 

TESTS: 

 

Check that the original error of :trac:`23891` is fixed:: 

 

sage: Permutations(4)([1,4,3,2]).weak_covers() 

[[1, 4, 2, 3], [1, 3, 4, 2]] 

""" 

if index_set is None: 

index_set = range(1, len(self)) 

 

if side == 'right': 

p = self 

else: 

p = self.inverse() 

descents = [] 

for i in index_set: 

if p[i-1] > p[i]: 

if not positive: 

descents.append(i) 

else: 

if positive: 

descents.append(i) 

 

if final_descent: 

descents.append(len(p)) 

 

if from_zero: 

return [i - 1 for i in descents] 

 

return descents 

 

def idescents(self, final_descent=False, from_zero=False): 

""" 

Return a list of the idescents of ``self``, that is the list of 

the descents of ``self``'s inverse. 

 

A descent of a permutation ``p`` is an integer ``i`` such that 

``p(i) > p(i+1)``. 

 

.. WARNING:: 

 

By default, the descents are returned as elements in the 

index set, i.e., starting at `1`. If you want them to 

start at `0`, set the keyword ``from_zero`` to ``True``. 

 

INPUT: 

 

- ``final_descent`` -- boolean (default ``False``); 

if ``True``, the last position of a non-empty 

permutation is also considered as a descent 

 

- ``from_zero`` -- optional boolean (default ``False``); 

if ``False``, return the positions starting from `1` 

 

EXAMPLES:: 

 

sage: Permutation([2,3,1]).idescents() 

[1] 

sage: Permutation([1,4,3,2]).idescents() 

[2, 3] 

sage: Permutation([1,4,3,2]).idescents(final_descent=True) 

[2, 3, 4] 

sage: Permutation([1,4,3,2]).idescents(from_zero=True) 

[1, 2] 

""" 

return self.inverse().descents(final_descent=final_descent, 

from_zero=from_zero) 

 

def idescents_signature(self, final_descent=False): 

""" 

Return the list obtained as follows: Each position in ``self`` 

is mapped to `-1` if it is an idescent and `1` if it is not an 

idescent. 

 

See :meth:`idescents` for a definition of idescents. 

 

With the ``final_descent`` option, the last position of a 

non-empty permutation is also considered as a descent. 

 

EXAMPLES:: 

 

sage: Permutation([1,4,3,2]).idescents() 

[2, 3] 

sage: Permutation([1,4,3,2]).idescents_signature() 

[1, -1, -1, 1] 

""" 

idescents = self.idescents(final_descent=final_descent) 

d = {True: -1, False: 1} 

return [d[(i + 1) in idescents] for i in range(len(self))] 

 

def number_of_descents(self, final_descent=False): 

r""" 

Return the number of descents of ``self``. 

 

With the ``final_descent`` option, the last position of a 

non-empty permutation is also considered as a descent. 

 

EXAMPLES:: 

 

sage: Permutation([1,4,3,2]).number_of_descents() 

2 

sage: Permutation([1,4,3,2]).number_of_descents(final_descent=True) 

3 

""" 

return len(self.descents(final_descent)) 

 

def number_of_idescents(self, final_descent=False): 

r""" 

Return the number of idescents of ``self``. 

 

See :meth:`idescents` for a definition of idescents. 

 

With the ``final_descent`` option, the last position of a 

non-empty permutation is also considered as a descent. 

 

EXAMPLES:: 

 

sage: Permutation([1,4,3,2]).number_of_idescents() 

2 

sage: Permutation([1,4,3,2]).number_of_idescents(final_descent=True) 

3 

""" 

return len(self.idescents(final_descent)) 

 

@combinatorial_map(name='descent composition') 

def descents_composition(self): 

r""" 

Return the descent composition of ``self``. 

 

The descent composition of a permutation `p \in S_n` is defined 

as the composition of `n` whose descent set equals the descent 

set of `p`. Here, the descent set of `p` is defined as the set 

of all `i \in \{ 1, 2, \ldots, n-1 \}` satisfying 

`p(i) > p(i+1)`. The descent set 

of a composition `c = (i_1, i_2, \ldots, i_k)` is defined as 

the set `\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, 

i_1 + i_2 + \cdots + i_{k-1} \}`. 

 

EXAMPLES:: 

 

sage: Permutation([1,3,2,4]).descents_composition() 

[2, 2] 

sage: Permutation([4,1,6,7,2,3,8,5]).descents_composition() 

[1, 3, 3, 1] 

sage: Permutation([]).descents_composition() 

[] 

""" 

if len(self) == 0: 

return Composition([]) 

d = [0] + self.descents(final_descent=True) 

return Composition([d[i + 1] - d[i] for i in range(len(d) - 1)]) 

 

def descent_polynomial(self): 

r""" 

Return the descent polynomial of the permutation ``self``. 

 

The descent polynomial of a permutation `p` is the product of 

all the ``z[p(i)]`` where ``i`` ranges over the descents of 

``p``. 

 

A descent of a permutation ``p`` is an integer ``i`` such that 

``p(i) > p(i+1)``. 

 

REFERENCES: 

 

.. [GarStan1984] \A. M. Garsia, Dennis Stanton. 

*Group actions on Stanley-Reisner rings and invariants of 

permutation groups*. Adv. in Math. **51** (1984), 107-201. 

http://www.sciencedirect.com/science/article/pii/0001870884900057 

 

EXAMPLES:: 

 

sage: Permutation([2,1,3]).descent_polynomial() 

z1 

sage: Permutation([4,3,2,1]).descent_polynomial() 

z1*z2^2*z3^3 

 

.. TODO:: 

 

This docstring needs to be fixed. First, the definition 

does not match the implementation (or the examples). 

Second, this doesn't seem to be defined in [GarStan1984]_ 

(the descent monomial in their (7.23) is different). 

""" 

p = self 

z = [] 

P = PolynomialRing(ZZ, len(p), 'z') 

z = P.gens() 

result = 1 

pol = 1 

for i in range(len(p)-1): 

pol *= z[p[i]-1] 

if p[i] > p[i+1]: 

result *= pol 

 

return result 

 

 

############## 

# Major Code # 

############## 

 

def major_index(self, final_descent=False): 

r""" 

Return the major index of ``self``. 

 

The major index of a permutation `p` is the sum of the descents of `p`. 

Since our permutation indices are 0-based, we need to add the 

number of descents. 

 

With the ``final_descent`` option, the last position of a 

non-empty permutation is also considered as a descent. 

 

EXAMPLES:: 

 

sage: Permutation([2,1,3]).major_index() 

1 

sage: Permutation([3,4,1,2]).major_index() 

2 

sage: Permutation([4,3,2,1]).major_index() 

6 

""" 

descents = self.descents(final_descent) 

return sum(descents) 

 

def imajor_index(self, final_descent=False): 

""" 

Return the inverse major index of the permutation ``self``, which is 

the major index of the inverse of ``self``. 

 

The major index of a permutation `p` is the sum of the descents of `p`. 

Since our permutation indices are 0-based, we need to add the 

number of descents. 

 

With the ``final_descent`` option, the last position of a 

non-empty permutation is also considered as a descent. 

 

EXAMPLES:: 

 

sage: Permutation([2,1,3]).imajor_index() 

1 

sage: Permutation([3,4,1,2]).imajor_index() 

2 

sage: Permutation([4,3,2,1]).imajor_index() 

6 

""" 

return sum(self.idescents(final_descent)) 

 

def to_major_code(self, final_descent=False): 

r""" 

Return the major code of the permutation ``self``. 

 

The major code of a permutation `p` is defined as the sequence 

`(m_1-m_2, m_2-m_3, \ldots, m_n)`, where `m_i` is the major 

index of the permutation obtained by erasing all letters smaller than 

`i` from `p`. 

 

With the ``final_descent`` option, the last position of a 

non-empty permutation is also considered as a descent. 

This has an effect on the computation of major indices. 

 

REFERENCES: 

 

- Carlitz, L. *q-Bernoulli and Eulerian Numbers*. 

Trans. Amer. Math. Soc. 76 (1954) 332-350. 

http://www.ams.org/journals/tran/1954-076-02/S0002-9947-1954-0060538-2/ 

 

- Skandera, M. *An Eulerian Partner for Inversions*. 

Sem. Lothar. Combin. 46 (2001) B46d. 

http://www.lehigh.edu/~mas906/papers/partner.ps 

 

EXAMPLES:: 

 

sage: Permutation([9,3,5,7,2,1,4,6,8]).to_major_code() 

[5, 0, 1, 0, 1, 2, 0, 1, 0] 

sage: Permutation([2,8,4,3,6,7,9,5,1]).to_major_code() 

[8, 3, 3, 1, 4, 0, 1, 0, 0] 

""" 

p = self 

n = len(p) 

major_indices = [0]*(n+1) 

smaller = p[:] 

P = Permutations() 

for i in range(n): 

major_indices[i] = P(smaller).major_index(final_descent) 

#Create the permutation that "erases" all the numbers 

#smaller than i+1 

smaller.remove(1) 

smaller = [i-1 for i in smaller] 

 

major_code = [ major_indices[i] - major_indices[i+1] for i in range(n) ] 

return major_code 

 

######### 

# Peaks # 

######### 

 

def peaks(self): 

r""" 

Return a list of the peaks of the permutation ``self``. 

 

A peak of a permutation `p` is an integer `i` such that 

`p(i-1) < p(i)` and `p(i) > p(i+1)`. 

 

EXAMPLES:: 

 

sage: Permutation([1,3,2,4,5]).peaks() 

[1] 

sage: Permutation([4,1,3,2,6,5]).peaks() 

[2, 4] 

sage: Permutation([]).peaks() 

[] 

""" 

p = self 

peaks = [] 

for i in range(1,len(p)-1): 

if p[i-1] <= p[i] and p[i] > p[i+1]: 

peaks.append(i) 

 

return peaks 

 

 

def number_of_peaks(self): 

r""" 

Return the number of peaks of the permutation ``self``. 

 

A peak of a permutation `p` is an integer `i` such that 

`p(i-1) < p(i)` and `p(i) > p(i+1)`. 

 

EXAMPLES:: 

 

sage: Permutation([1,3,2,4,5]).number_of_peaks() 

1 

sage: Permutation([4,1,3,2,6,5]).number_of_peaks() 

2 

""" 

return len(self.peaks()) 

 

############# 

# Saliances # 

############# 

 

def saliances(self): 

r""" 

Return a list of the saliances of the permutation ``self``. 

 

A saliance of a permutation `p` is an integer `i` such that 

`p(i) > p(j)` for all `j > i`. 

 

EXAMPLES:: 

 

sage: Permutation([2,3,1,5,4]).saliances() 

[3, 4] 

sage: Permutation([5,4,3,2,1]).saliances() 

[0, 1, 2, 3, 4] 

""" 

p = self 

saliances = [] 

for i in range(len(p)): 

is_saliance = True 

for j in range(i+1, len(p)): 

if p[i] <= p[j]: 

is_saliance = False 

if is_saliance: 

saliances.append(i) 

 

return saliances 

 

 

def number_of_saliances(self): 

r""" 

Return the number of saliances of ``self``. 

 

A saliance of a permutation `p` is an integer `i` such that 

`p(i) > p(j)` for all `j > i`. 

 

EXAMPLES:: 

 

sage: Permutation([2,3,1,5,4]).number_of_saliances() 

2 

sage: Permutation([5,4,3,2,1]).number_of_saliances() 

5 

""" 

return len(self.saliances()) 

 

################ 

# Bruhat Order # 

################ 

def bruhat_lequal(self, p2): 

r""" 

Return ``True`` if ``self`` is less or equal to ``p2`` in 

the Bruhat order. 

 

The Bruhat order (also called strong Bruhat order or Chevalley 

order) on the symmetric group `S_n` is the partial order on `S_n` 

determined by the following condition: If `p` is a permutation, 

and `i` and `j` are two indices satisfying `p(i) > p(j)` and 

`i < j` (that is, `(i, j)` is an inversion of `p` with `i < j`), 

then `p \circ (i, j)` (the permutation obtained by first 

switching `i` with `j` and then applying `p`) is smaller than `p` 

in the Bruhat order. 

 

One can show that a permutation `p \in S_n` is less or equal to 

a permutation `q \in S_n` in the Bruhat order if and only if 

for every `i \in \{ 0, 1, \cdots , n \}` and 

`j \in \{ 1, 2, \cdots , n \}`, the number of the elements among 

`p(1), p(2), \cdots, p(j)` that are greater than `i` is `\leq` 

to the number of the elements among `q(1), q(2), \cdots, q(j)` 

that are greater than `i`. 

 

This method assumes that ``self`` and ``p2`` are permutations 

of the same integer `n`. 

 

EXAMPLES:: 

 

sage: Permutation([2,4,3,1]).bruhat_lequal(Permutation([3,4,2,1])) 

True 

 

sage: Permutation([2,1,3]).bruhat_lequal(Permutation([2,3,1])) 

True 

sage: Permutation([2,1,3]).bruhat_lequal(Permutation([3,1,2])) 

True 

sage: Permutation([2,1,3]).bruhat_lequal(Permutation([1,2,3])) 

False 

sage: Permutation([1,3,2]).bruhat_lequal(Permutation([2,1,3])) 

False 

sage: Permutation([1,3,2]).bruhat_lequal(Permutation([2,3,1])) 

True 

sage: Permutation([2,3,1]).bruhat_lequal(Permutation([1,3,2])) 

False 

sage: sorted( [len([b for b in Permutations(3) if a.bruhat_lequal(b)]) 

....: for a in Permutations(3)] ) 

[1, 2, 2, 4, 4, 6] 

 

sage: Permutation([]).bruhat_lequal(Permutation([])) 

True 

""" 

p1 = self 

n1 = len(p1) 

 

if n1 == 0: 

return True 

 

if p1[0] > p2[0] or p1[n1-1] < p2[n1-1]: 

return False 

 

for i in range(1, n1): 

c = 0 

for j in range(n1 - 2): 

# We should really check this for all j in range(n1), but for 

# j == n1 - 1 it is tautological and for j == n1 - 2 the check 

# is contained in the check p1[n1-1] < p2[n1-1] already made. 

if p2[j] > i: 

c += 1 

if p1[j] > i: 

c -= 1 

if c < 0: 

return False 

 

return True 

 

def weak_excedences(self): 

""" 

Return all the numbers ``self[i]`` such that ``self[i] >= i+1``. 

 

EXAMPLES:: 

 

sage: Permutation([1,4,3,2,5]).weak_excedences() 

[1, 4, 3, 5] 

""" 

res = [] 

for i in range(len(self)): 

if self[i] >= i + 1: 

res.append(self[i]) 

return res 

 

 

def bruhat_inversions(self): 

r""" 

Return the list of inversions of ``self`` such that the application of 

this inversion to ``self`` decreases its number of inversions by 

exactly 1. 

 

Equivalently, it returns the list of pairs `(i,j)` such that `i < j`, 

such that `p(i) > p(j)` and such that there exists no `k` (strictly) 

between `i` and `j` satisfying `p(i) > p(k) > p(j)`. 

 

EXAMPLES:: 

 

sage: Permutation([5,2,3,4,1]).bruhat_inversions() 

[[0, 1], [0, 2], [0, 3], [1, 4], [2, 4], [3, 4]] 

sage: Permutation([6,1,4,5,2,3]).bruhat_inversions() 

[[0, 1], [0, 2], [0, 3], [2, 4], [2, 5], [3, 4], [3, 5]] 

""" 

return list(self.bruhat_inversions_iterator()) 

 

def bruhat_inversions_iterator(self): 

""" 

Return the iterator for the inversions of ``self`` such that the 

application of this inversion to ``self`` decreases its number of 

inversions by exactly 1. 

 

EXAMPLES:: 

 

sage: list(Permutation([5,2,3,4,1]).bruhat_inversions_iterator()) 

[[0, 1], [0, 2], [0, 3], [1, 4], [2, 4], [3, 4]] 

sage: list(Permutation([6,1,4,5,2,3]).bruhat_inversions_iterator()) 

[[0, 1], [0, 2], [0, 3], [2, 4], [2, 5], [3, 4], [3, 5]] 

""" 

p = self 

n = len(p) 

 

for i in range(n-1): 

for j in range(i+1,n): 

if p[i] > p[j]: 

ok = True 

for k in range(i+1, j): 

if p[i] > p[k] and p[k] > p[j]: 

ok = False 

break 

if ok: 

yield [i,j] 

 

 

def bruhat_succ(self): 

r""" 

Return a list of the permutations strictly greater than ``self`` in 

the Bruhat order (on the symmetric group containing ``self``) such 

that there is no permutation between one of those and ``self``. 

 

See :meth:`bruhat_lequal` for the definition of the Bruhat order. 

 

EXAMPLES:: 

 

sage: Permutation([6,1,4,5,2,3]).bruhat_succ() 

[[6, 4, 1, 5, 2, 3], 

[6, 2, 4, 5, 1, 3], 

[6, 1, 5, 4, 2, 3], 

[6, 1, 4, 5, 3, 2]] 

""" 

return list(self.bruhat_succ_iterator()) 

 

def bruhat_succ_iterator(self): 

""" 

An iterator for the permutations that are strictly greater than 

``self`` in the Bruhat order (on the symmetric group containing 

``self``) such that there is no permutation between one 

of those and ``self``. 

 

See :meth:`bruhat_lequal` for the definition of the Bruhat order. 

 

EXAMPLES:: 

 

sage: [x for x in Permutation([6,1,4,5,2,3]).bruhat_succ_iterator()] 

[[6, 4, 1, 5, 2, 3], 

[6, 2, 4, 5, 1, 3], 

[6, 1, 5, 4, 2, 3], 

[6, 1, 4, 5, 3, 2]] 

""" 

p = self 

n = len(p) 

P = Permutations() 

 

for z in P([n+1-x for x in p]).bruhat_inversions_iterator(): 

pp = p[:] 

pp[z[0]] = p[z[1]] 

pp[z[1]] = p[z[0]] 

yield P(pp) 

 

 

 

def bruhat_pred(self): 

r""" 

Return a list of the permutations strictly smaller than ``self`` 

in the Bruhat order (on the symmetric group containing ``self``) such 

that there is no permutation between one of those and ``self``. 

 

See :meth:`bruhat_lequal` for the definition of the Bruhat order. 

 

EXAMPLES:: 

 

sage: Permutation([6,1,4,5,2,3]).bruhat_pred() 

[[1, 6, 4, 5, 2, 3], 

[4, 1, 6, 5, 2, 3], 

[5, 1, 4, 6, 2, 3], 

[6, 1, 2, 5, 4, 3], 

[6, 1, 3, 5, 2, 4], 

[6, 1, 4, 2, 5, 3], 

[6, 1, 4, 3, 2, 5]] 

""" 

return list(self.bruhat_pred_iterator()) 

 

def bruhat_pred_iterator(self): 

""" 

An iterator for the permutations strictly smaller than ``self`` in 

the Bruhat order (on the symmetric group containing ``self``) such 

that there is no permutation between one of those and ``self``. 

 

See :meth:`bruhat_lequal` for the definition of the Bruhat order. 

 

EXAMPLES:: 

 

sage: [x for x in Permutation([6,1,4,5,2,3]).bruhat_pred_iterator()] 

[[1, 6, 4, 5, 2, 3], 

[4, 1, 6, 5, 2, 3], 

[5, 1, 4, 6, 2, 3], 

[6, 1, 2, 5, 4, 3], 

[6, 1, 3, 5, 2, 4], 

[6, 1, 4, 2, 5, 3], 

[6, 1, 4, 3, 2, 5]] 

""" 

p = self 

P = Permutations() 

for z in p.bruhat_inversions_iterator(): 

pp = p[:] 

pp[z[0]] = p[z[1]] 

pp[z[1]] = p[z[0]] 

yield P(pp) 

 

 

def bruhat_smaller(self): 

r""" 

Return the combinatorial class of permutations smaller than or 

equal to ``self`` in the Bruhat order (on the symmetric group 

containing ``self``). 

 

See :meth:`bruhat_lequal` for the definition of the Bruhat order. 

 

EXAMPLES:: 

 

sage: Permutation([4,1,2,3]).bruhat_smaller().list() 

[[1, 2, 3, 4], 

[1, 2, 4, 3], 

[1, 3, 2, 4], 

[1, 4, 2, 3], 

[2, 1, 3, 4], 

[2, 1, 4, 3], 

[3, 1, 2, 4], 

[4, 1, 2, 3]] 

""" 

return StandardPermutations_bruhat_smaller(self) 

 

 

def bruhat_greater(self): 

r""" 

Returns the combinatorial class of permutations greater than or 

equal to ``self`` in the Bruhat order (on the symmetric group 

containing ``self``). 

 

See :meth:`bruhat_lequal` for the definition of the Bruhat order. 

 

EXAMPLES:: 

 

sage: Permutation([4,1,2,3]).bruhat_greater().list() 

[[4, 1, 2, 3], 

[4, 1, 3, 2], 

[4, 2, 1, 3], 

[4, 2, 3, 1], 

[4, 3, 1, 2], 

[4, 3, 2, 1]] 

""" 

return StandardPermutations_bruhat_greater(self) 

 

######################## 

# Permutohedron Order # 

######################## 

 

def permutohedron_lequal(self, p2, side="right"): 

r""" 

Return ``True`` if ``self`` is less or equal to ``p2`` in the 

permutohedron order. 

 

By default, the computations are done in the right permutohedron. 

If you pass the option ``side='left'``, then they will be done in 

the left permutohedron. 

 

For every nonnegative integer `n`, the right (resp. left) 

permutohedron order (also called the right (resp. left) weak 

order, or the right (resp. left) weak Bruhat order) is a partial 

order on the symmetric group `S_n`. It can be defined in various 

ways, including the following ones: 

 

- Two permutations `u` and `v` in `S_n` satisfy `u \leq v` in 

the right (resp. left) permutohedron order if and only if 

the (Coxeter) length of the permutation `v^{-1} \circ u` 

(resp. of the permutation `u \circ v^{-1}`) equals the 

length of `v` minus the length of `u`. Here, `p \circ q` means 

the permutation obtained by applying `q` first and then `p`. 

(Recall that the Coxeter length of a permutation is its number 

of inversions.) 

 

- Two permutations `u` and `v` in `S_n` satisfy `u \leq v` in 

the right (resp. left) permutohedron order if and only if 

every pair `(i, j)` of elements of `\{ 1, 2, \cdots, n \}` 

such that `i < j` and `u^{-1}(i) > u^{-1}(j)` (resp. 

`u(i) > u(j)`) also satisfies `v^{-1}(i) > v^{-1}(j)` 

(resp. `v(i) > v(j)`). 

 

- A permutation `v \in S_n` covers a permutation `u \in S_n` in 

the right (resp. left) permutohedron order if and only if we 

have `v = u \circ (i, i + 1)` (resp. `v = (i, i + 1) \circ u`) 

for some `i \in \{ 1, 2, \cdots, n - 1 \}` satisfying 

`u(i) < u(i + 1)` (resp. `u^{-1}(i) < u^{-1}(i + 1)`). Here, 

again, `p \circ q` means the permutation obtained by applying 

`q` first and then `p`. 

 

The right and the left permutohedron order are mutually 

isomorphic, with the isomorphism being the map sending every 

permutation to its inverse. Each of these orders endows the 

symmetric group `S_n` with the structure of a graded poset 

(the rank function being the Coxeter length). 

 

.. WARNING:: 

 

The permutohedron order is not to be mistaken for the 

strong Bruhat order (:meth:`bruhat_lequal`), despite both 

orders being occasionally referred to as the Bruhat order. 

 

EXAMPLES:: 

 

sage: p = Permutation([3,2,1,4]) 

sage: p.permutohedron_lequal(Permutation([4,2,1,3])) 

False 

sage: p.permutohedron_lequal(Permutation([4,2,1,3]), side='left') 

True 

sage: p.permutohedron_lequal(p) 

True 

 

sage: Permutation([2,1,3]).permutohedron_lequal(Permutation([2,3,1])) 

True 

sage: Permutation([2,1,3]).permutohedron_lequal(Permutation([3,1,2])) 

False 

sage: Permutation([2,1,3]).permutohedron_lequal(Permutation([1,2,3])) 

False 

sage: Permutation([1,3,2]).permutohedron_lequal(Permutation([2,1,3])) 

False 

sage: Permutation([1,3,2]).permutohedron_lequal(Permutation([2,3,1])) 

False 

sage: Permutation([2,3,1]).permutohedron_lequal(Permutation([1,3,2])) 

False 

sage: Permutation([2,1,3]).permutohedron_lequal(Permutation([2,3,1]), side='left') 

False 

sage: sorted( [len([b for b in Permutations(3) if a.permutohedron_lequal(b)]) 

....: for a in Permutations(3)] ) 

[1, 2, 2, 3, 3, 6] 

sage: sorted( [len([b for b in Permutations(3) if a.permutohedron_lequal(b, side="left")]) 

....: for a in Permutations(3)] ) 

[1, 2, 2, 3, 3, 6] 

 

sage: Permutation([]).permutohedron_lequal(Permutation([])) 

True 

""" 

p1 = self 

l1 = p1.number_of_inversions() 

l2 = p2.number_of_inversions() 

 

if l1 > l2: 

return False 

 

if side == "right": 

prod = p1._left_to_right_multiply_on_right(p2.inverse()) 

else: 

prod = p1._left_to_right_multiply_on_left(p2.inverse()) 

 

return prod.number_of_inversions() == l2 - l1 

 

def permutohedron_succ(self, side="right"): 

r""" 

Return a list of the permutations strictly greater than ``self`` 

in the permutohedron order such that there is no permutation 

between any of those and ``self``. 

 

By default, the computations are done in the right permutohedron. 

If you pass the option ``side='left'``, then they will be done in 

the left permutohedron. 

 

See :meth:`permutohedron_lequal` for the definition of the 

permutohedron orders. 

 

EXAMPLES:: 

 

sage: p = Permutation([4,2,1,3]) 

sage: p.permutohedron_succ() 

[[4, 2, 3, 1]] 

sage: p.permutohedron_succ(side='left') 

[[4, 3, 1, 2]] 

""" 

p = self 

n = len(p) 

P = Permutations() 

succ = [] 

if side == "right": 

rise = lambda perm: [i for i in range(0,n-1) if perm[i] < perm[i+1]] 

for i in rise(p): 

pp = p[:] 

pp[i] = p[i+1] 

pp[i+1] = p[i] 

succ.append(P(pp)) 

else: 

advance = lambda perm: [i for i in range(1,n) if perm.index(i) < perm.index(i+1)] 

for i in advance(p): 

pp = p[:] 

pp[p.index(i)] = i+1 

pp[p.index(i+1)] = i 

succ.append(P(pp)) 

return succ 

 

 

def permutohedron_pred(self, side="right"): 

r""" 

Return a list of the permutations strictly smaller than ``self`` 

in the permutohedron order such that there is no permutation 

between any of those and ``self``. 

 

By default, the computations are done in the right permutohedron. 

If you pass the option ``side='left'``, then they will be done in 

the left permutohedron. 

 

See :meth:`permutohedron_lequal` for the definition of the 

permutohedron orders. 

 

EXAMPLES:: 

 

sage: p = Permutation([4,2,1,3]) 

sage: p.permutohedron_pred() 

[[2, 4, 1, 3], [4, 1, 2, 3]] 

sage: p.permutohedron_pred(side='left') 

[[4, 1, 2, 3], [3, 2, 1, 4]] 

""" 

p = self 

n = len(p) 

P = Permutations() 

pred = [] 

if side == "right": 

for d in p.descents(): 

pp = p[:] 

pp[d - 1] = p[d] 

pp[d] = p[d - 1] 

pred.append(P(pp)) 

else: 

recoil = lambda perm: [i for i in range(1,n) if perm.index(i) > perm.index(i+1)] 

for i in recoil(p): 

pp = p[:] 

pp[p.index(i)] = i+1 

pp[p.index(i+1)] = i 

pred.append(P(pp)) 

return pred 

 

 

def permutohedron_smaller(self, side="right"): 

r""" 

Return a list of permutations smaller than or equal to ``self`` 

in the permutohedron order. 

 

By default, the computations are done in the right permutohedron. 

If you pass the option ``side='left'``, then they will be done in 

the left permutohedron. 

 

See :meth:`permutohedron_lequal` for the definition of the 

permutohedron orders. 

 

EXAMPLES:: 

 

sage: Permutation([4,2,1,3]).permutohedron_smaller() 

[[1, 2, 3, 4], 

[1, 2, 4, 3], 

[1, 4, 2, 3], 

[2, 1, 3, 4], 

[2, 1, 4, 3], 

[2, 4, 1, 3], 

[4, 1, 2, 3], 

[4, 2, 1, 3]] 

 

:: 

 

sage: Permutation([4,2,1,3]).permutohedron_smaller(side='left') 

[[1, 2, 3, 4], 

[1, 3, 2, 4], 

[2, 1, 3, 4], 

[2, 3, 1, 4], 

[3, 1, 2, 4], 

[3, 2, 1, 4], 

[4, 1, 2, 3], 

[4, 2, 1, 3]] 

""" 

return transitive_ideal(lambda x: x.permutohedron_pred(side), self) 

 

 

def permutohedron_greater(self, side="right"): 

r""" 

Return a list of permutations greater than or equal to ``self`` 

in the permutohedron order. 

 

By default, the computations are done in the right permutohedron. 

If you pass the option ``side='left'``, then they will be done in 

the left permutohedron. 

 

See :meth:`permutohedron_lequal` for the definition of the 

permutohedron orders. 

 

EXAMPLES:: 

 

sage: Permutation([4,2,1,3]).permutohedron_greater() 

[[4, 2, 1, 3], [4, 2, 3, 1], [4, 3, 2, 1]] 

sage: Permutation([4,2,1,3]).permutohedron_greater(side='left') 

[[4, 2, 1, 3], [4, 3, 1, 2], [4, 3, 2, 1]] 

""" 

return transitive_ideal(lambda x: x.permutohedron_succ(side), self) 

 

def right_permutohedron_interval_iterator(self, other): 

r""" 

Return an iterator on the permutations (represented as integer 

lists) belonging to the right permutohedron interval where 

``self`` is the minimal element and ``other`` the maximal element. 

 

See :meth:`permutohedron_lequal` for the definition of the 

permutohedron orders. 

 

EXAMPLES:: 

 

sage: Permutation([2, 1, 4, 5, 3]).right_permutohedron_interval(Permutation([2, 5, 4, 1, 3])) # indirect doctest 

[[2, 1, 4, 5, 3], [2, 1, 5, 4, 3], [2, 4, 1, 5, 3], [2, 4, 5, 1, 3], [2, 5, 1, 4, 3], [2, 5, 4, 1, 3]] 

""" 

if len(self) != len(other) : 

raise ValueError("len({}) and len({}) must be equal".format(self, other)) 

if not self.permutohedron_lequal(other) : 

raise ValueError("{} must be lower or equal than {} for the right permutohedron order".format(self, other)) 

from sage.graphs.linearextensions import LinearExtensions 

d = DiGraph() 

d.add_vertices(range(1, len(self) + 1)) 

d.add_edges([(j, i) for (i, j) in self.inverse().inversions()]) 

d.add_edges([(other[i], other[j]) for i in range(len(other) - 1) 

for j in range(i, len(other)) if other[i] < other[j]]) 

return LinearExtensions(d) 

 

def right_permutohedron_interval(self, other): 

r""" 

Return the list of the permutations belonging to the right 

permutohedron interval where ``self`` is the minimal element and 

``other`` the maximal element. 

 

See :meth:`permutohedron_lequal` for the definition of the 

permutohedron orders. 

 

EXAMPLES:: 

 

sage: Permutation([2, 1, 4, 5, 3]).right_permutohedron_interval(Permutation([2, 5, 4, 1, 3])) 

[[2, 1, 4, 5, 3], [2, 1, 5, 4, 3], [2, 4, 1, 5, 3], [2, 4, 5, 1, 3], [2, 5, 1, 4, 3], [2, 5, 4, 1, 3]] 

 

TESTS:: 

 

sage: Permutation([]).right_permutohedron_interval(Permutation([])) 

[[]] 

sage: Permutation([3, 1, 2]).right_permutohedron_interval(Permutation([3, 1, 2])) 

[[3, 1, 2]] 

sage: Permutation([1, 3, 2, 4]).right_permutohedron_interval(Permutation([3, 4, 2, 1])) 

[[1, 3, 2, 4], [1, 3, 4, 2], [3, 1, 2, 4], [3, 1, 4, 2], [3, 2, 1, 4], [3, 2, 4, 1], [3, 4, 1, 2], [3, 4, 2, 1]] 

sage: Permutation([2, 1, 4, 5, 3]).right_permutohedron_interval(Permutation([2, 5, 4, 1, 3])) 

[[2, 1, 4, 5, 3], [2, 1, 5, 4, 3], [2, 4, 1, 5, 3], [2, 4, 5, 1, 3], [2, 5, 1, 4, 3], [2, 5, 4, 1, 3]] 

sage: Permutation([2, 5, 4, 1, 3]).right_permutohedron_interval(Permutation([2, 1, 4, 5, 3])) 

Traceback (most recent call last): 

... 

ValueError: [2, 5, 4, 1, 3] must be lower or equal than [2, 1, 4, 5, 3] for the right permutohedron order 

sage: Permutation([2, 4, 1, 3]).right_permutohedron_interval(Permutation([2, 1, 4, 5, 3])) 

Traceback (most recent call last): 

... 

ValueError: len([2, 4, 1, 3]) and len([2, 1, 4, 5, 3]) must be equal 

""" 

P = Permutations() 

return [P(p) for p in self.right_permutohedron_interval_iterator(other)] 

 

def permutohedron_join(self, other, side="right"): 

r""" 

Return the join of the permutations ``self`` and ``other`` 

in the right permutohedron order (or, if ``side`` is set to 

``'left'``, in the left permutohedron order). 

 

The permutohedron orders (see :meth:`permutohedron_lequal`) 

are lattices; the join operation refers to this lattice 

structure. In more elementary terms, the join of two 

permutations `\pi` and `\psi` in the symmetric group `S_n` 

is the permutation in `S_n` whose set of inversion is the 

transitive closure of the union of the set of inversions of 

`\pi` with the set of inversions of `\psi`. 

 

.. SEEALSO:: 

 

:meth:`permutohedron_lequal`, :meth:`permutohedron_meet`. 

 

ALGORITHM: 

 

It is enough to construct the join of any two permutations 

`\pi` and `\psi` in `S_n` with respect to the right weak 

order. (The join of `\pi` and `\psi` with respect to the 

left weak order is the inverse of the join of `\pi^{-1}` 

and `\psi^{-1}` with respect to the right weak order.) 

Start with an empty list `l` (denoted ``xs`` in the actual 

code). For `i = 1, 2, \ldots, n` (in this order), we insert 

`i` into this list in the rightmost possible position such 

that any letter in `\{ 1, 2, ..., i-1 \}` which appears 

further right than `i` in either `\pi` or `\psi` (or both) 

must appear further right than `i` in the resulting list. 

After all numbers are inserted, we are left with a list 

which is precisely the join of `\pi` and `\psi` (in 

one-line notation). This algorithm is due to Markowsky, 

[Mark94]_ (Theorem 1 (a)). 

 

REFERENCES: 

 

.. [Mark94] George Markowsky. 

*Permutation lattices revisited*. 

Mathematical Social Sciences, 27 (1994), 59--72. 

 

AUTHORS: 

 

Viviane Pons and Darij Grinberg, 18 June 2014. 

 

EXAMPLES:: 

 

sage: p = Permutation([3,1,2]) 

sage: q = Permutation([1,3,2]) 

sage: p.permutohedron_join(q) 

[3, 1, 2] 

sage: r = Permutation([2,1,3]) 

sage: r.permutohedron_join(p) 

[3, 2, 1] 

 

:: 

 

sage: p = Permutation([3,2,4,1]) 

sage: q = Permutation([4,2,1,3]) 

sage: p.permutohedron_join(q) 

[4, 3, 2, 1] 

sage: r = Permutation([3,1,2,4]) 

sage: p.permutohedron_join(r) 

[3, 2, 4, 1] 

sage: q.permutohedron_join(r) 

[4, 3, 2, 1] 

sage: s = Permutation([1,4,2,3]) 

sage: s.permutohedron_join(r) 

[4, 3, 1, 2] 

 

The universal property of the join operation is 

satisfied:: 

 

sage: def test_uni_join(p, q): 

....: j = p.permutohedron_join(q) 

....: if not p.permutohedron_lequal(j): 

....: return False 

....: if not q.permutohedron_lequal(j): 

....: return False 

....: for r in p.permutohedron_greater(): 

....: if q.permutohedron_lequal(r) and not j.permutohedron_lequal(r): 

....: return False 

....: return True 

sage: all( test_uni_join(p, q) for p in Permutations(3) for q in Permutations(3) ) 

True 

sage: test_uni_join(Permutation([6, 4, 7, 3, 2, 5, 8, 1]), Permutation([7, 3, 1, 2, 5, 4, 6, 8])) 

True 

 

Border cases:: 

 

sage: p = Permutation([]) 

sage: p.permutohedron_join(p) 

[] 

sage: p = Permutation([1]) 

sage: p.permutohedron_join(p) 

[1] 

 

The left permutohedron: 

 

sage: p = Permutation([3,1,2]) 

sage: q = Permutation([1,3,2]) 

sage: p.permutohedron_join(q, side="left") 

[3, 2, 1] 

sage: r = Permutation([2,1,3]) 

sage: r.permutohedron_join(p, side="left") 

[3, 1, 2] 

""" 

if side == "left": 

return self.inverse().permutohedron_join(other.inverse()).inverse() 

n = self.size() 

xs = [] 

for i in range(1, n + 1): 

u = self.index(i) 

must_be_right = [f for f in self[u + 1:] if f < i] 

v = other.index(i) 

must_be_right += [f for f in other[v + 1:] if f < i] 

must_be_right = uniq(sorted(must_be_right)) 

for j, q in enumerate(xs): 

if q in must_be_right: 

xs = xs[:j] + [i] + xs[j:] 

break 

else: 

xs.append(i) 

return Permutations(n)(xs) 

 

def permutohedron_meet(self, other, side="right"): 

r""" 

Return the meet of the permutations ``self`` and ``other`` 

in the right permutohedron order (or, if ``side`` is set to 

``'left'``, in the left permutohedron order). 

 

The permutohedron orders (see :meth:`permutohedron_lequal`) 

are lattices; the meet operation refers to this lattice 

structure. It is connected to the join operation by the 

following simple symmetry property: If `\pi` and `\psi` 

are two permutations `\pi` and `\psi` in the symmetric group 

`S_n`, and if `w_0` denotes the permutation 

`(n, n-1, \ldots, 1) \in S_n`, then 

 

.. MATH:: 

 

\pi \wedge \psi = w_0 \circ ((w_0 \circ \pi) \vee (w_0 \circ \psi)) 

= ((\pi \circ w_0) \vee (\psi \circ w_0)) \circ w_0 

 

and 

 

.. MATH:: 

 

\pi \vee \psi = w_0 \circ ((w_0 \circ \pi) \wedge (w_0 \circ \psi)) 

= ((\pi \circ w_0) \wedge (\psi \circ w_0)) \circ w_0, 

 

where `\wedge` means meet and `\vee` means join. 

 

.. SEEALSO:: 

 

:meth:`permutohedron_lequal`, :meth:`permutohedron_join`. 

 

AUTHORS: 

 

Viviane Pons and Darij Grinberg, 18 June 2014. 

 

EXAMPLES:: 

 

sage: p = Permutation([3,1,2]) 

sage: q = Permutation([1,3,2]) 

sage: p.permutohedron_meet(q) 

[1, 3, 2] 

sage: r = Permutation([2,1,3]) 

sage: r.permutohedron_meet(p) 

[1, 2, 3] 

 

:: 

 

sage: p = Permutation([3,2,4,1]) 

sage: q = Permutation([4,2,1,3]) 

sage: p.permutohedron_meet(q) 

[2, 1, 3, 4] 

sage: r = Permutation([3,1,2,4]) 

sage: p.permutohedron_meet(r) 

[3, 1, 2, 4] 

sage: q.permutohedron_meet(r) 

[1, 2, 3, 4] 

sage: s = Permutation([1,4,2,3]) 

sage: s.permutohedron_meet(r) 

[1, 2, 3, 4] 

 

The universal property of the meet operation is 

satisfied:: 

 

sage: def test_uni_meet(p, q): 

....: m = p.permutohedron_meet(q) 

....: if not m.permutohedron_lequal(p): 

....: return False 

....: if not m.permutohedron_lequal(q): 

....: return False 

....: for r in p.permutohedron_smaller(): 

....: if r.permutohedron_lequal(q) and not r.permutohedron_lequal(m): 

....: return False 

....: return True 

sage: all( test_uni_meet(p, q) for p in Permutations(3) for q in Permutations(3) ) 

True 

sage: test_uni_meet(Permutation([6, 4, 7, 3, 2, 5, 8, 1]), Permutation([7, 3, 1, 2, 5, 4, 6, 8])) 

True 

 

Border cases:: 

 

sage: p = Permutation([]) 

sage: p.permutohedron_meet(p) 

[] 

sage: p = Permutation([1]) 

sage: p.permutohedron_meet(p) 

[1] 

 

The left permutohedron: 

 

sage: p = Permutation([3,1,2]) 

sage: q = Permutation([1,3,2]) 

sage: p.permutohedron_meet(q, side="left") 

[1, 2, 3] 

sage: r = Permutation([2,1,3]) 

sage: r.permutohedron_meet(p, side="left") 

[2, 1, 3] 

""" 

return self.reverse().permutohedron_join(other.reverse(), side=side).reverse() 

 

############ 

# Patterns # 

############ 

 

def has_pattern(self, patt): 

r""" 

Test whether the permutation ``self`` contains the pattern 

``patt``. 

 

EXAMPLES:: 

 

sage: Permutation([3,5,1,4,6,2]).has_pattern([1,3,2]) 

True 

""" 

p = self 

n = len(p) 

l = len(patt) 

if l > n: 

return False 

for pos in itertools.combinations(range(n), l): 

if to_standard([p[z] for z in pos]) == patt: 

return True 

return False 

 

def avoids(self, patt): 

""" 

Test whether the permutation ``self`` avoids the pattern 

``patt``. 

 

EXAMPLES:: 

 

sage: Permutation([6,2,5,4,3,1]).avoids([4,2,3,1]) 

False 

sage: Permutation([6,1,2,5,4,3]).avoids([4,2,3,1]) 

True 

sage: Permutation([6,1,2,5,4,3]).avoids([3,4,1,2]) 

True 

""" 

return not self.has_pattern(patt) 

 

def pattern_positions(self, patt): 

r""" 

Return the list of positions where the pattern ``patt`` appears 

in the permutation ``self``. 

 

EXAMPLES:: 

 

sage: Permutation([3,5,1,4,6,2]).pattern_positions([1,3,2]) 

[[0, 1, 3], [2, 3, 5], [2, 4, 5]] 

""" 

p = self 

 

return [list(pos) for pos in itertools.combinations(range(len(p)), len(patt)) 

if to_standard([p[z] for z in pos]) == patt] 

 

@combinatorial_map(name='Simion-Schmidt map') 

def simion_schmidt(self, avoid=[1,2,3]): 

r""" 

Implements the Simion-Schmidt map which sends an arbitrary permutation 

to a pattern avoiding permutation, where the permutation pattern is one 

of four length-three patterns. This method also implements the bijection 

between (for example) ``[1,2,3]``- and ``[1,3,2]``-avoiding permutations. 

 

INPUT: 

 

- ``avoid`` -- one of the patterns ``[1,2,3]``, ``[1,3,2]``, ``[3,1,2]``, ``[3,2,1]``. 

 

EXAMPLES:: 

 

sage: P=Permutations(6) 

sage: p=P([4,5,1,6,3,2]) 

sage: pl= [ [1,2,3], [1,3,2], [3,1,2], [3,2,1] ] 

sage: for q in pl: 

....: s=p.simion_schmidt(q) 

....: print("{} {}".format(s, s.has_pattern(q))) 

[4, 6, 1, 5, 3, 2] False 

[4, 2, 1, 3, 5, 6] False 

[4, 5, 3, 6, 2, 1] False 

[4, 5, 1, 6, 2, 3] False 

""" 

if len(list(self)) <= 2: 

return self 

targetPermutation = [self[0]] 

extreme = self[0] 

nonMinima = [] 

if avoid == [1,2,3] or avoid == [1,3,2]: 

for i in range(1, len(list(self))): 

if self[i] < extreme: 

targetPermutation.append(self[i]) 

extreme = self[i] 

else: 

targetPermutation.append(None) 

nonMinima.append(self[i]) 

nonMinima.sort() 

if avoid == [1,3,2]: 

nonMinima.reverse() 

if avoid == [3,2,1] or avoid == [3,1,2]: 

for i in range(1, len(list(self))): 

if self[i] > extreme: 

targetPermutation.append(self[i]) 

extreme = self[i] 

else: 

targetPermutation.append(None) 

nonMinima.append(self[i]) 

nonMinima.sort() 

if avoid == [3,2,1]: 

nonMinima.reverse() 

 

for i in range(1, len(list(self))): 

if targetPermutation[i] is None: 

targetPermutation[i] = nonMinima.pop() 

return Permutations()(targetPermutation) 

 

@combinatorial_map(order=2,name='reverse') 

def reverse(self): 

""" 

Returns the permutation obtained by reversing the list. 

 

EXAMPLES:: 

 

sage: Permutation([3,4,1,2]).reverse() 

[2, 1, 4, 3] 

sage: Permutation([1,2,3,4,5]).reverse() 

[5, 4, 3, 2, 1] 

""" 

return self.__class__(self.parent(), [i for i in reversed(self)] ) 

 

@combinatorial_map(order=2,name='complement') 

def complement(self): 

r""" 

Return the complement of the permutation ``self``. 

 

The complement of a permutation `w \in S_n` is defined as the 

permutation in `S_n` sending each `i` to `n + 1 - w(i)`. 

 

EXAMPLES:: 

 

sage: Permutation([1,2,3]).complement() 

[3, 2, 1] 

sage: Permutation([1, 3, 2]).complement() 

[3, 1, 2] 

""" 

n = len(self) 

return self.__class__(self.parent(), [n - x + 1 for x in self] ) 

 

@combinatorial_map(name='permutation poset') 

def permutation_poset(self): 

r""" 

Return the permutation poset of ``self``. 

 

The permutation poset of a permutation `p` is the poset with 

vertices `(i, p(i))` for `i = 1, 2, \ldots, n` (where `n` is the 

size of `p`) and order inherited from `\ZZ \times \ZZ`. 

 

EXAMPLES:: 

 

sage: Permutation([3,1,5,4,2]).permutation_poset().cover_relations() 

[[(2, 1), (5, 2)], 

[(2, 1), (3, 5)], 

[(2, 1), (4, 4)], 

[(1, 3), (3, 5)], 

[(1, 3), (4, 4)]] 

sage: Permutation([]).permutation_poset().cover_relations() 

[] 

sage: Permutation([1,3,2]).permutation_poset().cover_relations() 

[[(1, 1), (2, 3)], [(1, 1), (3, 2)]] 

sage: Permutation([1,2]).permutation_poset().cover_relations() 

[[(1, 1), (2, 2)]] 

sage: P = Permutation([1,5,2,4,3]) 

sage: P.permutation_poset().greene_shape() == P.RS_partition() # This should hold for any P. 

True 

""" 

from sage.combinat.posets.posets import Poset 

n = len(self) 

posetdict = {} 

for i in range(n): 

u = self[i] 

posetdict[(i + 1, u)] = [(j + 1, self[j]) for j in range(i + 1, n) if u < self[j]] 

return Poset(posetdict) 

 

def dict(self): 

""" 

Returns a dictionary corresponding to the permutation. 

 

EXAMPLES:: 

 

sage: p = Permutation([2,1,3]) 

sage: d = p.dict() 

sage: d[1] 

2 

sage: d[2] 

1 

sage: d[3] 

3 

""" 

d = {} 

for i in range(len(self)): 

d[i+1] = self[i] 

return d 

 

def action(self, a): 

""" 

Return the action of the permutation ``self`` on a list ``a``. 

 

The action of a permutation `p \in S_n` on an `n`-element list 

`(a_1, a_2, \ldots, a_n)` is defined to be 

`(a_{p(1)}, a_{p(2)}, \ldots, a_{p(n)})`. 

 

EXAMPLES:: 

 

sage: p = Permutation([2,1,3]) 

sage: a = list(range(3)) 

sage: p.action(a) 

[1, 0, 2] 

sage: b = [1,2,3,4] 

sage: p.action(b) 

Traceback (most recent call last): 

... 

ValueError: len(a) must equal len(self) 

 

sage: q = Permutation([2,3,1]) 

sage: a = list(range(3)) 

sage: q.action(a) 

[1, 2, 0] 

""" 

if len(a) != len(self): 

raise ValueError("len(a) must equal len(self)") 

return [a[i-1] for i in self] 

 

###################### 

# Robinson-Schensted # 

###################### 

 

def robinson_schensted(self): 

""" 

Return the pair of standard tableaux obtained by running the 

Robinson-Schensted algorithm on ``self``. 

 

This can also be done by running 

:func:`~sage.combinat.rsk.RSK` on ``self`` (with the optional argument 

``check_standard=True`` to return standard Young tableaux). 

 

EXAMPLES:: 

 

sage: Permutation([6,2,3,1,7,5,4]).robinson_schensted() 

[[[1, 3, 4], [2, 5], [6, 7]], [[1, 3, 5], [2, 6], [4, 7]]] 

""" 

return RSK(self, check_standard=True) 

 

def _rsk_iter(self): 

r""" 

An iterator for RSK. 

 

Yields pairs ``[i, p(i)]`` for a permutation ``p``. 

 

EXAMPLES:: 

 

sage: for x in Permutation([6,2,3,1,7,5,4])._rsk_iter(): x 

... 

(1, 6) 

(2, 2) 

(3, 3) 

(4, 1) 

(5, 7) 

(6, 5) 

(7, 4) 

""" 

return zip(range(1, len(self) + 1), self) 

 

@combinatorial_map(name='Robinson-Schensted insertion tableau') 

def left_tableau(self): 

""" 

Return the left standard tableau after performing the RSK 

algorithm on ``self``. 

 

EXAMPLES:: 

 

sage: Permutation([1,4,3,2]).left_tableau() 

[[1, 2], [3], [4]] 

""" 

return RSK(self, check_standard=True)[0] 

 

@combinatorial_map(name='Robinson-Schensted recording tableau') 

def right_tableau(self): 

""" 

Return the right standard tableau after performing the RSK 

algorithm on ``self``. 

 

EXAMPLES:: 

 

sage: Permutation([1,4,3,2]).right_tableau() 

[[1, 2], [3], [4]] 

""" 

return RSK(self, check_standard=True)[1] 

 

def increasing_tree(self, compare=min): 

""" 

Return the increasing tree associated to ``self``. 

 

EXAMPLES:: 

 

sage: Permutation([1,4,3,2]).increasing_tree() 

1[., 2[3[4[., .], .], .]] 

sage: Permutation([4,1,3,2]).increasing_tree() 

1[4[., .], 2[3[., .], .]] 

 

By passing the option ``compare=max`` one can have the decreasing 

tree instead:: 

 

sage: Permutation([2,3,4,1]).increasing_tree(max) 

4[3[2[., .], .], 1[., .]] 

sage: Permutation([2,3,1,4]).increasing_tree(max) 

4[3[2[., .], 1[., .]], .] 

""" 

from sage.combinat.binary_tree import LabelledBinaryTree as LBT 

def rec(perm): 

if len(perm) == 0: return LBT(None) 

mn = compare(perm) 

k = perm.index(mn) 

return LBT([rec(perm[:k]), rec(perm[k+1:])], label = mn) 

return rec(self) 

 

@combinatorial_map(name="Increasing tree") 

def increasing_tree_shape(self, compare=min): 

r""" 

Return the shape of the increasing tree associated with the 

permutation. 

 

EXAMPLES:: 

 

sage: Permutation([1,4,3,2]).increasing_tree_shape() 

[., [[[., .], .], .]] 

sage: Permutation([4,1,3,2]).increasing_tree_shape() 

[[., .], [[., .], .]] 

 

By passing the option ``compare=max`` one can have the decreasing 

tree instead:: 

 

sage: Permutation([2,3,4,1]).increasing_tree_shape(max) 

[[[., .], .], [., .]] 

sage: Permutation([2,3,1,4]).increasing_tree_shape(max) 

[[[., .], [., .]], .] 

""" 

return self.increasing_tree(compare).shape() 

 

def binary_search_tree(self, left_to_right=True): 

""" 

Return the binary search tree associated to ``self``. 

 

If `w` is a word, then the binary search tree associated to `w` 

is defined as the result of starting with an empty binary tree, 

and then inserting the letters of `w` one by one into this tree. 

Here, the insertion is being done according to the method 

:meth:`~sage.combinat.binary_tree.LabelledBinaryTree.binary_search_insert`, 

and the word `w` is being traversed from left to right. 

 

A permutation is regarded as a word (using one-line notation), 

and thus a binary search tree associated to a permutation is 

defined. 

 

If the optional keyword variable ``left_to_right`` is set to 

``False``, the word `w` is being traversed from right to left 

instead. 

 

EXAMPLES:: 

 

sage: Permutation([1,4,3,2]).binary_search_tree() 

1[., 4[3[2[., .], .], .]] 

sage: Permutation([4,1,3,2]).binary_search_tree() 

4[1[., 3[2[., .], .]], .] 

 

By passing the option ``left_to_right=False`` one can have 

the insertion going from right to left:: 

 

sage: Permutation([1,4,3,2]).binary_search_tree(False) 

2[1[., .], 3[., 4[., .]]] 

sage: Permutation([4,1,3,2]).binary_search_tree(False) 

2[1[., .], 3[., 4[., .]]] 

 

TESTS:: 

 

sage: Permutation([]).binary_search_tree() 

. 

""" 

from sage.combinat.binary_tree import LabelledBinaryTree as LBT 

res = LBT(None) 

if left_to_right: 

gen = self 

else: 

gen = self[::-1] 

for i in gen: 

res = res.binary_search_insert(i) 

return res 

 

@combinatorial_map(name = "Binary search tree (left to right)") 

def binary_search_tree_shape(self, left_to_right=True): 

r""" 

Return the shape of the binary search tree of the permutation 

(a non labelled binary tree). 

 

EXAMPLES:: 

 

sage: Permutation([1,4,3,2]).binary_search_tree_shape() 

[., [[[., .], .], .]] 

sage: Permutation([4,1,3,2]).binary_search_tree_shape() 

[[., [[., .], .]], .] 

 

By passing the option ``left_to_right=False`` one can have 

the insertion going from right to left:: 

 

sage: Permutation([1,4,3,2]).binary_search_tree_shape(False) 

[[., .], [., [., .]]] 

sage: Permutation([4,1,3,2]).binary_search_tree_shape(False) 

[[., .], [., [., .]]] 

""" 

from sage.combinat.binary_tree import binary_search_tree_shape 

return binary_search_tree_shape(list(self), left_to_right) 

 

def sylvester_class(self, left_to_right=False): 

""" 

Iterate over the equivalence class of the permutation ``self`` 

under sylvester congruence. 

 

Sylvester congruence is an equivalence relation on the set `S_n` 

of all permutations of `n`. It is defined as the smallest 

equivalence relation such that every permutation of the form 

`uacvbw` with `u`, `v` and `w` being words and `a`, `b` and `c` 

being letters satisfying `a \leq b < c` is equivalent to the 

permutation `ucavbw`. (Here, permutations are regarded as words 

by way of one-line notation.) This definition comes from [HNT05]_, 

Definition 8, where it is more generally applied to arbitrary 

words. 

 

The equivalence class of a permutation `p \in S_n` under sylvester 

congruence is called the *sylvester class* of `p`. It is an 

interval in the right permutohedron order (see 

:meth:`permutohedron_lequal`) on `S_n`. 

 

This is related to the 

:meth:`~sage.combinat.binary_tree.LabelledBinaryTree.sylvester_class` 

method in that the equivalence class of a permutation `\pi` under 

sylvester congruence is the sylvester class of the right-to-left 

binary search tree of `\pi`. However, the present method 

yields permutations, while the method on labelled binary trees 

yields plain lists. 

 

If the variable ``left_to_right`` is set to ``True``, the method 

instead iterates over the equivalence class of ``self`` with 

respect to the *left* sylvester congruence. The left sylvester 

congruence is easiest to define by saying that two permutations 

are equivalent under it if and only if their reverses 

(:meth:`reverse`) are equivalent under (standard) sylvester 

congruence. 

 

EXAMPLES: 

 

The sylvester class of a permutation in `S_5`:: 

 

sage: p = Permutation([3, 5, 1, 2, 4]) 

sage: sorted(p.sylvester_class()) 

[[1, 3, 2, 5, 4], 

[1, 3, 5, 2, 4], 

[1, 5, 3, 2, 4], 

[3, 1, 2, 5, 4], 

[3, 1, 5, 2, 4], 

[3, 5, 1, 2, 4], 

[5, 1, 3, 2, 4], 

[5, 3, 1, 2, 4]] 

 

The sylvester class of a permutation `p` contains `p`:: 

 

sage: all( p in p.sylvester_class() for p in Permutations(4) ) 

True 

 

Small cases:: 

 

sage: list(Permutation([]).sylvester_class()) 

[[]] 

 

sage: list(Permutation([1]).sylvester_class()) 

[[1]] 

 

The sylvester classes in `S_3`:: 

 

sage: [sorted(p.sylvester_class()) for p in Permutations(3)] 

[[[1, 2, 3]], 

[[1, 3, 2], [3, 1, 2]], 

[[2, 1, 3]], 

[[2, 3, 1]], 

[[1, 3, 2], [3, 1, 2]], 

[[3, 2, 1]]] 

 

The left sylvester classes in `S_3`:: 

 

sage: [sorted(p.sylvester_class(left_to_right=True)) for p in Permutations(3)] 

[[[1, 2, 3]], 

[[1, 3, 2]], 

[[2, 1, 3], [2, 3, 1]], 

[[2, 1, 3], [2, 3, 1]], 

[[3, 1, 2]], 

[[3, 2, 1]]] 

 

A left sylvester class in `S_5`:: 

 

sage: p = Permutation([4, 2, 1, 5, 3]) 

sage: sorted(p.sylvester_class(left_to_right=True)) 

[[4, 2, 1, 3, 5], 

[4, 2, 1, 5, 3], 

[4, 2, 3, 1, 5], 

[4, 2, 3, 5, 1], 

[4, 2, 5, 1, 3], 

[4, 2, 5, 3, 1], 

[4, 5, 2, 1, 3], 

[4, 5, 2, 3, 1]] 

""" 

parself = self.parent() 

t = self.binary_search_tree(left_to_right=left_to_right) 

for u in t.sylvester_class(left_to_right=left_to_right): 

yield parself(u) 

 

@combinatorial_map(name='Robinson-Schensted tableau shape') 

def RS_partition(self): 

""" 

Return the shape of the tableaux obtained by applying the RSK 

algorithm to ``self``. 

 

EXAMPLES:: 

 

sage: Permutation([1,4,3,2]).RS_partition() 

[2, 1, 1] 

""" 

return RSK(self)[1].shape() 

 

def remove_extra_fixed_points(self): 

""" 

Return the permutation obtained by removing any fixed points at 

the end of ``self``. 

 

EXAMPLES:: 

 

sage: Permutation([2,1,3]).remove_extra_fixed_points() 

[2, 1] 

sage: Permutation([1,2,3,4]).remove_extra_fixed_points() 

[1] 

 

.. SEEALSO:: 

 

:meth:`retract_plain` 

""" 

#Strip off all extra fixed points at the end of 

#the permutation. 

i = len(self)-1 

while i >= 1: 

if i != self[i] - 1: 

break 

i -= 1 

return Permutations()(self[:i+1]) 

 

def retract_plain(self, m): 

r""" 

Return the plain retract of the permutation ``self`` in `S_n` 

to `S_m`, where `m \leq n`. If this retract is undefined, then 

``None`` is returned. 

 

If `p \in S_n` is a permutation, and `m` is a nonnegative integer 

less or equal to `n`, then the plain retract of `p` to `S_m` is 

defined only if every `i > m` satisfies `p(i) = i`. In this case, 

it is defined as the permutation written 

`(p(1), p(2), \ldots, p(m))` in one-line notation. 

 

EXAMPLES:: 

 

sage: Permutation([4,1,2,3,5]).retract_plain(4) 

[4, 1, 2, 3] 

sage: Permutation([4,1,2,3,5]).retract_plain(3) 

 

sage: Permutation([1,3,2,4,5,6]).retract_plain(3) 

[1, 3, 2] 

sage: Permutation([1,3,2,4,5,6]).retract_plain(2) 

 

sage: Permutation([1,2,3,4,5]).retract_plain(1) 

[1] 

sage: Permutation([1,2,3,4,5]).retract_plain(0) 

[] 

 

sage: all( p.retract_plain(3) == p for p in Permutations(3) ) 

True 

 

.. SEEALSO:: 

 

:meth:`retract_direct_product`, :meth:`retract_okounkov_vershik`, 

:meth:`remove_extra_fixed_points` 

""" 

n = len(self) 

p = list(self) 

for i in range(m, n): 

if p[i] != i + 1: 

return None 

return Permutations(m)(p[:m]) 

 

def retract_direct_product(self, m): 

r""" 

Return the direct-product retract of the permutation 

``self`` `\in S_n` to `S_m`, where `m \leq n`. If this retract 

is undefined, then ``None`` is returned. 

 

If `p \in S_n` is a permutation, and `m` is a nonnegative integer 

less or equal to `n`, then the direct-product retract of `p` to 

`S_m` is defined only if `p([m]) = [m]`, where `[m]` denotes the 

interval `\{1, 2, \ldots, m\}`. In this case, it is defined as the 

permutation written `(p(1), p(2), \ldots, p(m))` in one-line 

notation. 

 

EXAMPLES:: 

 

sage: Permutation([4,1,2,3,5]).retract_direct_product(4) 

[4, 1, 2, 3] 

sage: Permutation([4,1,2,3,5]).retract_direct_product(3) 

 

sage: Permutation([1,4,2,3,6,5]).retract_direct_product(5) 

sage: Permutation([1,4,2,3,6,5]).retract_direct_product(4) 

[1, 4, 2, 3] 

sage: Permutation([1,4,2,3,6,5]).retract_direct_product(3) 

sage: Permutation([1,4,2,3,6,5]).retract_direct_product(2) 

sage: Permutation([1,4,2,3,6,5]).retract_direct_product(1) 

[1] 

sage: Permutation([1,4,2,3,6,5]).retract_direct_product(0) 

[] 

 

sage: all( p.retract_direct_product(3) == p for p in Permutations(3) ) 

True 

 

.. SEEALSO:: 

 

:meth:`retract_plain`, :meth:`retract_okounkov_vershik` 

""" 

n = len(self) 

p = list(self) 

for i in range(m, n): 

if p[i] <= m: 

return None 

return Permutations(m)(p[:m]) 

 

def retract_okounkov_vershik(self, m): 

r""" 

Return the Okounkov-Vershik retract of the permutation 

``self`` `\in S_n` to `S_m`, where `m \leq n`. 

 

If `p \in S_n` is a permutation, and `m` is a nonnegative integer 

less or equal to `n`, then the Okounkov-Vershik retract of `p` to 

`S_m` is defined as the permutation in `S_m` which sends every 

`i \in \{1, 2, \ldots, m\}` to `p^{k_i}(i)`, where `k_i` is the 

smallest positive integer `k` satisfying `p^k(i) \leq m`. 

 

In other words, the Okounkov-Vershik retract of `p` is the 

permutation whose disjoint cycle decomposition is obtained by 

removing all letters strictly greater than `m` from the 

decomposition of `p` into disjoint cycles (and removing all 

cycles which are emptied in the process). 

 

When `m = n-1`, the Okounkov-Vershik retract (as a map 

`S_n \to S_{n-1}`) is the map `\widetilde{p}_n` introduced in 

Section 7 of [OkounkovVershik2]_, and appears as (3.20) in 

[CST10]_. In the general case, the Okounkov-Vershik retract 

of a permutation in `S_n` to `S_m` can be obtained by first 

taking its Okounkov-Vershik retract to `S_{n-1}`, then that 

of the resulting permutation to `S_{n-2}`, etc. until arriving 

in `S_m`. 

 

REFERENCES: 

 

.. [OkounkovVershik2] \A. M. Vershik, A. Yu. Okounkov. 

*A New Approach to the Representation Theory of the Symmetric 

Groups. 2*. :arxiv:`math/0503040v3`. 

 

.. [CST10] Tullio Ceccherini-Silberstein, Fabio Scarabotti, 

Filippo Tolli. 

*Representation Theory of the Symmetric Groups: The 

Okounkov-Vershik Approach, Character Formulas, and Partition 

Algebras*. CUP 2010. 

 

EXAMPLES:: 

 

sage: Permutation([4,1,2,3,5]).retract_okounkov_vershik(4) 

[4, 1, 2, 3] 

sage: Permutation([4,1,2,3,5]).retract_okounkov_vershik(3) 

[3, 1, 2] 

sage: Permutation([4,1,2,3,5]).retract_okounkov_vershik(2) 

[2, 1] 

sage: Permutation([4,1,2,3,5]).retract_okounkov_vershik(1) 

[1] 

sage: Permutation([4,1,2,3,5]).retract_okounkov_vershik(0) 

[] 

 

sage: Permutation([1,4,2,3,6,5]).retract_okounkov_vershik(5) 

[1, 4, 2, 3, 5] 

sage: Permutation([1,4,2,3,6,5]).retract_okounkov_vershik(4) 

[1, 4, 2, 3] 

sage: Permutation([1,4,2,3,6,5]).retract_okounkov_vershik(3) 

[1, 3, 2] 

sage: Permutation([1,4,2,3,6,5]).retract_okounkov_vershik(2) 

[1, 2] 

sage: Permutation([1,4,2,3,6,5]).retract_okounkov_vershik(1) 

[1] 

sage: Permutation([1,4,2,3,6,5]).retract_okounkov_vershik(0) 

[] 

 

sage: Permutation([6,5,4,3,2,1]).retract_okounkov_vershik(5) 

[1, 5, 4, 3, 2] 

sage: Permutation([6,5,4,3,2,1]).retract_okounkov_vershik(4) 

[1, 2, 4, 3] 

 

sage: Permutation([1,5,2,6,3,7,4,8]).retract_okounkov_vershik(4) 

[1, 3, 2, 4] 

 

sage: all( p.retract_direct_product(3) == p for p in Permutations(3) ) 

True 

 

.. SEEALSO:: 

 

:meth:`retract_plain`, :meth:`retract_direct_product` 

""" 

res = [] 

for i in range(1, m + 1): 

j = self(i) 

while j > m: 

j = self(j) 

res.append(j) 

return Permutations(m)(res) 

 

def hyperoctahedral_double_coset_type(self): 

r""" 

Return the coset-type of ``self`` as a partition. 

 

``self`` must be a permutation of even size `2n`. The coset-type 

determines the double class of the permutation, that is its image in 

`H_n \backslash S_{2n} / H_n`, where `H_n` is the `n`-th 

hyperoctahedral group. 

 

The coset-type is determined as follows. Consider the perfect matching 

`\{\{1,2\},\{3,4\},\dots,\{2n-1,2n\}\}` and its image by ``self``, and 

draw them simultaneously as edges of a graph whose vertices are labeled 

by `1,2,\dots,2n`. The coset-type is the ordered sequence of the 

semi-lengths of the cycles of this graph (see Chapter VII of [Mcd]_ for 

more details, particularly Section VII.2). 

 

EXAMPLES:: 

 

sage: Permutation([3, 4, 6, 1, 5, 7, 2, 8]).hyperoctahedral_double_coset_type() 

[3, 1] 

sage: all(p.hyperoctahedral_double_coset_type() == 

....: p.inverse().hyperoctahedral_double_coset_type() 

....: for p in Permutations(4)) 

True 

sage: Permutation([]).hyperoctahedral_double_coset_type() 

[] 

sage: Permutation([3,1,2]).hyperoctahedral_double_coset_type() 

Traceback (most recent call last): 

... 

ValueError: [3, 1, 2] is a permutation of odd size and has no coset-type 

 

REFERENCES: 

 

.. [Mcd] \I. G. Macdonald. Symmetric functions and Hall 

polynomials. Oxford University Press, second edition, 1995. 

""" 

from sage.combinat.perfect_matching import PerfectMatchings 

n = len(self) 

if n % 2 == 1: 

raise ValueError("%s is a permutation of odd size and has no coset-type"%self) 

S = PerfectMatchings(n)([(2*i+1,2*i+2) for i in range(n//2)]) 

return S.loop_type(S.apply_permutation(self)) 

 

##################### 

# Binary operations # 

##################### 

 

def shifted_concatenation(self, other, side = "right"): 

r""" 

Return the right (or left) shifted concatenation of ``self`` 

with a permutation ``other``. These operations are also known 

as the Loday-Ronco over and under operations. 

 

INPUT: 

 

- ``other`` -- a permutation, a list, a tuple, or any iterable 

representing a permutation. 

 

- ``side`` -- (default: ``"right"``) the string "left" or "right". 

 

OUTPUT: 

 

If ``side`` is ``"right"``, the method returns the permutation 

obtained by concatenating ``self`` with the letters of ``other`` 

incremented by the size of ``self``. This is what is called 

``side / other`` in [LodRon0102066]_, and denoted as the "over" 

operation. 

Otherwise, i. e., when ``side`` is ``"left"``, the method 

returns the permutation obtained by concatenating the letters 

of ``other`` incremented by the size of ``self`` with ``self``. 

This is what is called ``side \ other`` in [LodRon0102066]_ 

(which seems to use the `(\sigma \pi)(i) = \pi(\sigma(i))` 

convention for the product of permutations). 

 

EXAMPLES:: 

 

sage: Permutation([]).shifted_concatenation(Permutation([]), "right") 

[] 

sage: Permutation([]).shifted_concatenation(Permutation([]), "left") 

[] 

sage: Permutation([2, 4, 1, 3]).shifted_concatenation(Permutation([3, 1, 2]), "right") 

[2, 4, 1, 3, 7, 5, 6] 

sage: Permutation([2, 4, 1, 3]).shifted_concatenation(Permutation([3, 1, 2]), "left") 

[7, 5, 6, 2, 4, 1, 3] 

 

REFERENCES: 

 

.. [LodRon0102066] Jean-Louis Loday and Maria O. Ronco. 

Order structure on the algebra of permutations 

and of planar binary trees. 

:arXiv:`math/0102066v1`. 

""" 

if side == "right" : 

return Permutations()(list(self) + [a + len(self) for a in other]) 

elif side == "left" : 

return Permutations()([a + len(self) for a in other] + list(self)) 

else : 

raise ValueError("%s must be \"left\" or \"right\"" %(side)) 

 

def shifted_shuffle(self, other): 

r""" 

Return the shifted shuffle of two permutations ``self`` and ``other``. 

 

INPUT: 

 

- ``other`` -- a permutation, a list, a tuple, or any iterable 

representing a permutation. 

 

OUTPUT: 

 

The list of the permutations appearing in the shifted 

shuffle of the permutations ``self`` and ``other``. 

 

EXAMPLES:: 

 

sage: Permutation([]).shifted_shuffle(Permutation([])) 

[[]] 

sage: Permutation([1, 2, 3]).shifted_shuffle(Permutation([1])) 

[[1, 2, 3, 4], [1, 2, 4, 3], [1, 4, 2, 3], [4, 1, 2, 3]] 

sage: Permutation([1, 2]).shifted_shuffle(Permutation([2, 1])) 

[[1, 2, 4, 3], [1, 4, 2, 3], [1, 4, 3, 2], [4, 1, 2, 3], [4, 1, 3, 2], [4, 3, 1, 2]] 

sage: Permutation([1]).shifted_shuffle([1]) 

[[1, 2], [2, 1]] 

sage: len(Permutation([3, 1, 5, 4, 2]).shifted_shuffle(Permutation([2, 1, 4, 3]))) 

126 

 

The shifted shuffle product is associative. We can test this on an 

admittedly toy example:: 

 

sage: all( all( all( sorted(flatten([abs.shifted_shuffle(c) 

....: for abs in a.shifted_shuffle(b)])) 

....: == sorted(flatten([a.shifted_shuffle(bcs) 

....: for bcs in b.shifted_shuffle(c)])) 

....: for c in Permutations(2) ) 

....: for b in Permutations(2) ) 

....: for a in Permutations(2) ) 

True 

 

The ``shifted_shuffle`` method on permutations gives the same 

permutations as the ``shifted_shuffle`` method on words (but is 

faster):: 

 

sage: all( all( sorted(p1.shifted_shuffle(p2)) 

....: == sorted([Permutation(p) for p in 

....: Word(p1).shifted_shuffle(Word(p2))]) 

....: for p2 in Permutations(3) ) 

....: for p1 in Permutations(2) ) 

True 

""" 

return self.shifted_concatenation(other, "right").\ 

right_permutohedron_interval(self.shifted_concatenation(other, "left")) 

 

################################################################ 

# Parent classes 

################################################################ 

 

# Base class for permutations 

class Permutations(UniqueRepresentation, Parent): 

r""" 

Permutations. 

 

``Permutations(n)`` returns the class of permutations of ``n``, if ``n`` 

is an integer, list, set, or string. 

 

``Permutations(n, k)`` returns the class of length-``k`` partial 

permutations of ``n`` (where ``n`` is any of the above things); ``k`` 

must be a nonnegative integer. A length-`k` partial permutation of `n` 

is defined as a `k`-tuple of pairwise distinct elements of 

`\{ 1, 2, \ldots, n \}`. 

 

Valid keyword arguments are: 'descents', 'bruhat_smaller', 

'bruhat_greater', 'recoils_finer', 'recoils_fatter', 'recoils', 

and 'avoiding'. With the exception of 'avoiding', you cannot 

specify ``n`` or ``k`` along with a keyword. 

 

``Permutations(descents=(list,n))`` returns the class of permutations of 

`n` with descents in the positions specified by ``list``. This uses the 

slightly nonstandard convention that the images of `1,2,...,n` under the 

permutation are regarded as positions `0,1,...,n-1`, so for example the 

presence of `1` in ``list`` signifies that the permutations `\pi` should 

satisfy `\pi(2) > \pi(3)`. 

Note that ``list`` is supposed to be a list of positions of the descents, 

not the descents composition. It does *not* return the class of 

permutations with descents composition ``list``. 

 

``Permutations(bruhat_smaller=p)`` and ``Permutations(bruhat_greater=p)`` 

return the class of permutations smaller-or-equal or greater-or-equal, 

respectively, than the given permutation ``p`` in the Bruhat order. 

(The Bruhat order is defined in 

:meth:`~sage.combinat.permutation.Permutation.bruhat_lequal`. 

It is also referred to as the *strong* Bruhat order.) 

 

``Permutations(recoils=p)`` returns the class of permutations whose 

recoils composition is ``p``. Unlike the ``descents=(list, n)`` syntax, 

this actually takes a *composition* as input. 

 

``Permutations(recoils_fatter=p)`` and ``Permutations(recoils_finer=p)`` 

return the class of permutations whose recoils composition is fatter or 

finer, respectively, than the given composition ``p``. 

 

``Permutations(n, avoiding=P)`` returns the class of permutations of ``n`` 

avoiding ``P``. Here ``P`` may be a single permutation or a list of 

permutations; the returned class will avoid all patterns in ``P``. 

 

EXAMPLES:: 

 

sage: p = Permutations(3); p 

Standard permutations of 3 

sage: p.list() 

[[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]] 

 

:: 

 

sage: p = Permutations(3, 2); p 

Permutations of {1,...,3} of length 2 

sage: p.list() 

[[1, 2], [1, 3], [2, 1], [2, 3], [3, 1], [3, 2]] 

 

:: 

 

sage: p = Permutations(['c', 'a', 't']); p 

Permutations of the set ['c', 'a', 't'] 

sage: p.list() 

[['c', 'a', 't'], 

['c', 't', 'a'], 

['a', 'c', 't'], 

['a', 't', 'c'], 

['t', 'c', 'a'], 

['t', 'a', 'c']] 

 

:: 

 

sage: p = Permutations(['c', 'a', 't'], 2); p 

Permutations of the set ['c', 'a', 't'] of length 2 

sage: p.list() 

[['c', 'a'], ['c', 't'], ['a', 'c'], ['a', 't'], ['t', 'c'], ['t', 'a']] 

 

:: 

 

sage: p = Permutations([1,1,2]); p 

Permutations of the multi-set [1, 1, 2] 

sage: p.list() 

[[1, 1, 2], [1, 2, 1], [2, 1, 1]] 

 

:: 

 

sage: p = Permutations([1,1,2], 2); p 

Permutations of the multi-set [1, 1, 2] of length 2 

sage: p.list() 

[[1, 1], [1, 2], [2, 1]] 

 

:: 

 

sage: p = Permutations(descents=([1], 4)); p 

Standard permutations of 4 with descents [1] 

sage: p.list() 

[[1, 3, 2, 4], [1, 4, 2, 3], [2, 3, 1, 4], [2, 4, 1, 3], [3, 4, 1, 2]] 

 

:: 

 

sage: p = Permutations(bruhat_smaller=[1,3,2,4]); p 

Standard permutations that are less than or equal to [1, 3, 2, 4] in the Bruhat order 

sage: p.list() 

[[1, 2, 3, 4], [1, 3, 2, 4]] 

 

:: 

 

sage: p = Permutations(bruhat_greater=[4,2,3,1]); p 

Standard permutations that are greater than or equal to [4, 2, 3, 1] in the Bruhat order 

sage: p.list() 

[[4, 2, 3, 1], [4, 3, 2, 1]] 

 

:: 

 

sage: p = Permutations(recoils_finer=[2,1]); p 

Standard permutations whose recoils composition is finer than [2, 1] 

sage: p.list() 

[[1, 2, 3], [1, 3, 2], [3, 1, 2]] 

 

:: 

 

sage: p = Permutations(recoils_fatter=[2,1]); p 

Standard permutations whose recoils composition is fatter than [2, 1] 

sage: p.list() 

[[1, 3, 2], [3, 1, 2], [3, 2, 1]] 

 

:: 

 

sage: p = Permutations(recoils=[2,1]); p 

Standard permutations whose recoils composition is [2, 1] 

sage: p.list() 

[[1, 3, 2], [3, 1, 2]] 

 

:: 

 

sage: p = Permutations(4, avoiding=[1,3,2]); p 

Standard permutations of 4 avoiding [1, 3, 2] 

sage: p.list() 

[[4, 1, 2, 3], 

[4, 2, 1, 3], 

[4, 2, 3, 1], 

[4, 3, 1, 2], 

[4, 3, 2, 1], 

[3, 4, 1, 2], 

[3, 4, 2, 1], 

[2, 3, 4, 1], 

[3, 2, 4, 1], 

[1, 2, 3, 4], 

[2, 1, 3, 4], 

[2, 3, 1, 4], 

[3, 1, 2, 4], 

[3, 2, 1, 4]] 

 

:: 

 

sage: p = Permutations(5, avoiding=[[3,4,1,2], [4,2,3,1]]); p 

Standard permutations of 5 avoiding [[3, 4, 1, 2], [4, 2, 3, 1]] 

sage: p.cardinality() 

88 

sage: p.random_element() 

[5, 1, 2, 4, 3] 

""" 

@staticmethod 

def __classcall_private__(cls, n=None, k=None, **kwargs): 

""" 

Return the correct parent based upon input. 

 

EXAMPLES:: 

 

sage: Permutations() 

Standard permutations 

sage: Permutations(5, 3) 

Permutations of {1,...,5} of length 3 

sage: Permutations([1,2,3,4,6]) 

Permutations of the set [1, 2, 3, 4, 6] 

sage: Permutations([1,2,3,4,5]) 

Standard permutations of 5 

""" 

valid_args = ['descents', 'bruhat_smaller', 'bruhat_greater', 

'recoils_finer', 'recoils_fatter', 'recoils', 'avoiding'] 

 

number_of_arguments = 0 

if n is not None: 

number_of_arguments += 1 

elif k is not None: 

number_of_arguments += 1 

 

#Make sure that exactly one keyword was passed 

for key in kwargs: 

if key not in valid_args: 

raise ValueError("unknown keyword argument: %s"%key) 

if key not in [ 'avoiding' ]: 

number_of_arguments += 1 

 

if number_of_arguments == 0: 

return StandardPermutations_all() 

 

if number_of_arguments != 1: 

raise ValueError("you must specify exactly one argument") 

 

if n is not None: 

if isinstance(n, (int, Integer)): 

if k is None: 

if 'avoiding' in kwargs: 

a = kwargs['avoiding'] 

if a in StandardPermutations_all(): 

if a == [1,2]: 

return StandardPermutations_avoiding_12(n) 

elif a == [2,1]: 

return StandardPermutations_avoiding_21(n) 

elif a == [1,2,3]: 

return StandardPermutations_avoiding_123(n) 

elif a == [1,3,2]: 

return StandardPermutations_avoiding_132(n) 

elif a == [2,1,3]: 

return StandardPermutations_avoiding_213(n) 

elif a == [2,3,1]: 

return StandardPermutations_avoiding_231(n) 

elif a == [3,1,2]: 

return StandardPermutations_avoiding_312(n) 

elif a == [3,2,1]: 

return StandardPermutations_avoiding_321(n) 

else: 

return StandardPermutations_avoiding_generic(n, (a,)) 

elif isinstance(a, (list, tuple)): 

a = tuple(map(Permutation, a)) 

return StandardPermutations_avoiding_generic(n, a) 

else: 

raise ValueError("do not know how to avoid %s"%a) 

else: 

return StandardPermutations_n(n) 

else: 

return Permutations_nk(n,k) 

else: 

# In this case, we have that n is a list 

# Because of UniqueRepresentation, we require the elements 

# to be hashable 

if len(set(n)) == len(n): 

if list(n) == list(range(1, len(n)+1)): 

if k is None: 

return StandardPermutations_n(len(n)) 

else: 

return Permutations_nk(len(n), k) 

else: 

if k is None: 

return Permutations_set(n) 

else: 

return Permutations_setk(n,k) 

else: 

if k is None: 

return Permutations_mset(n) 

else: 

return Permutations_msetk(n,k) 

elif 'descents' in kwargs: 

#Descent positions specified 

if isinstance(kwargs['descents'], tuple): 

#Descent positions and size specified 

args = kwargs['descents'] 

return StandardPermutations_descents(tuple(args[0]), args[1]) 

else: 

#Size not specified 

return StandardPermutations_descents(kwargs['descents']) 

elif 'bruhat_smaller' in kwargs: 

return StandardPermutations_bruhat_smaller(Permutation(kwargs['bruhat_smaller'])) 

elif 'bruhat_greater' in kwargs: 

return StandardPermutations_bruhat_greater(Permutation(kwargs['bruhat_greater'])) 

elif 'recoils_finer' in kwargs: 

return StandardPermutations_recoilsfiner(Composition(kwargs['recoils_finer'])) 

elif 'recoils_fatter' in kwargs: 

return StandardPermutations_recoilsfatter(Composition(kwargs['recoils_fatter'])) 

elif 'recoils' in kwargs: 

return StandardPermutations_recoils(Composition(kwargs['recoils'])) 

 

Element = Permutation 

 

# add options to class 

class options(GlobalOptions): 

r""" 

Set the global options for elements of the permutation class. The 

defaults are for permutations to be displayed in list notation and 

the multiplication done from left to right (like in GAP) -- that 

is, `(\pi \psi)(i) = \psi(\pi(i))` for all `i`. 

 

.. NOTE:: 

 

These options have no effect on permutation group elements. 

 

@OPTIONS@ 

 

EXAMPLES:: 

 

sage: p213 = Permutation([2,1,3]) 

sage: p312 = Permutation([3,1,2]) 

sage: Permutations.options(mult='l2r', display='list') 

sage: Permutations.options.display 

list 

sage: p213 

[2, 1, 3] 

sage: Permutations.options.display='cycle' 

sage: p213 

(1,2) 

sage: Permutations.options.display='singleton' 

sage: p213 

(1,2)(3) 

sage: Permutations.options.display='list' 

 

:: 

 

sage: Permutations.options.mult 

l2r 

sage: p213*p312 

[1, 3, 2] 

sage: Permutations.options.mult='r2l' 

sage: p213*p312 

[3, 2, 1] 

sage: Permutations.options._reset() 

""" 

NAME = 'Permutations' 

module = 'sage.combinat.permutation' 

display = dict(default="list", 

description="Specifies how the permutations should be printed", 

values=dict(list="the permutations are displayed in list notation" 

" (aka 1-line notation)", 

cycle="the permutations are displayed in cycle notation" 

" (i. e., as products of disjoint cycles)", 

singleton="the permutations are displayed in cycle notation" 

" with singleton cycles shown as well", 

reduced_word="the permutations are displayed as reduced words"), 

alias=dict(word="reduced_word", reduced_expression="reduced_word"), 

case_sensitive=False) 

latex = dict(default="list", 

description="Specifies how the permutations should be latexed", 

values=dict(list="latex as a list in one-line notation", 

twoline="latex in two-line notation", 

cycle="latex in cycle notation", 

singleton="latex in cycle notation with singleton cycles shown as well", 

reduced_word="latex as reduced words"), 

alias=dict(word="reduced_word", reduced_expression="reduced_word", oneline="list"), 

case_sensitive=False) 

latex_empty_str = dict(default="1", 

description='The LaTeX representation of a reduced word when said word is empty', 

checker=lambda char: isinstance(char,str)) 

generator_name = dict(default="s", 

description="the letter used in latexing the reduced word", 

checker=lambda char: isinstance(char,str)) 

mult = dict(default="l2r", 

description="The multiplication of permutations", 

values=dict(l2r="left to right: `(p_1 \cdot p_2)(x) = p_2(p_1(x))`", 

r2l="right to left: `(p_1 \cdot p_2)(x) = p_1(p_2(x))`"), 

case_sensitive=False) 

 

class Permutations_nk(Permutations): 

r""" 

Length-`k` partial permutations of `\{1, 2, \ldots, n\}`. 

""" 

def __init__(self, n, k): 

""" 

TESTS:: 

 

sage: P = Permutations(3,2) 

sage: TestSuite(P).run() 

""" 

self.n = n 

self.k = k 

Permutations.__init__(self, category=FiniteEnumeratedSets()) 

 

class Element(ClonableArray): 

""" 

A length-`k` partial permutation of `[n]`. 

""" 

def check(self): 

""" 

Verify that ``self`` is a valid length-`k` partial 

permutation of `[n]`. 

 

EXAMPLES:: 

 

sage: S = Permutations(4, 2) 

sage: elt = S([3, 1]) 

sage: elt.check() 

""" 

if self not in self.parent(): 

raise ValueError("Invalid permutation") 

 

def __contains__(self, x): 

""" 

EXAMPLES:: 

 

sage: [1,2] in Permutations(3,2) 

True 

sage: [1,1] in Permutations(3,2) 

False 

sage: [3,2,1] in Permutations(3,2) 

False 

sage: [3,1] in Permutations(3,2) 

True 

""" 

if len(x) != self.k: return False 

 

r = list(range(1, self.n+1)) 

for i in x: 

if i in r: 

r.remove(i) 

else: 

return False 

 

return True 

 

def _repr_(self): 

""" 

TESTS:: 

 

sage: Permutations(3,2) 

Permutations of {1,...,3} of length 2 

""" 

return "Permutations of {1,...,%s} of length %s" % (self.n, self.k) 

 

def __iter__(self): 

""" 

EXAMPLES:: 

 

sage: [p for p in Permutations(3,2)] 

[[1, 2], [1, 3], [2, 1], [2, 3], [3, 1], [3, 2]] 

sage: [p for p in Permutations(3,0)] 

[[]] 

sage: [p for p in Permutations(3,4)] 

[] 

""" 

for x in itertools.permutations(range(1,self.n+1), int(self.k)): 

yield self.element_class(self, x) 

 

def cardinality(self): 

""" 

EXAMPLES:: 

 

sage: Permutations(3,0).cardinality() 

1 

sage: Permutations(3,1).cardinality() 

3 

sage: Permutations(3,2).cardinality() 

6 

sage: Permutations(3,3).cardinality() 

6 

sage: Permutations(3,4).cardinality() 

0 

""" 

if self.k <= self.n and self.k >= 0: 

return factorial(self.n) // factorial(self.n-self.k) 

return ZZ.zero() 

 

def random_element(self): 

""" 

EXAMPLES:: 

 

sage: Permutations(3,2).random_element() 

[1, 2] 

""" 

return sample(range(1, self.n+1), self.k) 

 

class Permutations_mset(Permutations): 

r""" 

Permutations of a multiset `M`. 

 

A permutation of a multiset `M` is represented by a list that 

contains exactly the same elements as `M` (with the same 

multiplicities), but possibly in different order. If `M` is 

a proper set there are `|M| !` such permutations. 

Otherwise, if the first element appears `k_1` times, the 

second element appears `k_2` times and so on, the number 

of permutations is `|M|! / (k_1! k_2! \ldots)`, which 

is sometimes called a multinomial coefficient. 

 

EXAMPLES:: 

 

sage: mset = [1,1,2,2,2] 

sage: from sage.combinat.permutation import Permutations_mset 

sage: P = Permutations_mset(mset); P 

Permutations of the multi-set [1, 1, 2, 2, 2] 

sage: sorted(P) 

[[1, 1, 2, 2, 2], 

[1, 2, 1, 2, 2], 

[1, 2, 2, 1, 2], 

[1, 2, 2, 2, 1], 

[2, 1, 1, 2, 2], 

[2, 1, 2, 1, 2], 

[2, 1, 2, 2, 1], 

[2, 2, 1, 1, 2], 

[2, 2, 1, 2, 1], 

[2, 2, 2, 1, 1]] 

sage: MS = MatrixSpace(GF(2),2,2) 

sage: A = MS([1,0,1,1]) 

sage: rows = A.rows() 

sage: rows[0].set_immutable() 

sage: rows[1].set_immutable() 

sage: P = Permutations_mset(rows); P 

Permutations of the multi-set [(1, 0), (1, 1)] 

sage: sorted(P) 

[[(1, 0), (1, 1)], [(1, 1), (1, 0)]] 

""" 

@staticmethod 

def __classcall_private__(cls, mset): 

""" 

Normalize arguments to ensure a unique representation. 

 

TESTS:: 

 

sage: S1 = Permutations(['c','a','c']) 

sage: S2 = Permutations(('c','a','c')) 

sage: S1 is S2 

True 

""" 

return super(Permutations_mset, cls).__classcall__(cls, tuple(mset)) 

 

def __init__(self, mset): 

""" 

TESTS:: 

 

sage: S = Permutations(['c','a','c']) 

sage: TestSuite(S).run() 

""" 

self.mset = mset 

Permutations.__init__(self, category=FiniteEnumeratedSets()) 

 

def __contains__(self, x): 

""" 

EXAMPLES:: 

 

sage: p = Permutations([1,2,2]) 

sage: [1,2,2] in p 

True 

sage: [] in p 

False 

sage: [2,2] in p 

False 

sage: [1,1] in p 

False 

sage: [2,1] in p 

False 

sage: [2,1,2] in p 

True 

""" 

s = list(self.mset) 

if len(x) != len(s): 

return False 

for i in x: 

if i in s: 

s.remove(i) 

else: 

return False 

return True 

 

class Element(ClonableArray): 

""" 

A permutation of an arbitrary multiset. 

""" 

def check(self): 

""" 

Verify that ``self`` is a valid permutation of the underlying 

multiset. 

 

EXAMPLES:: 

 

sage: S = Permutations(['c','a','c']) 

sage: elt = S(['c','c','a']) 

sage: elt.check() 

""" 

if self not in self.parent(): 

raise ValueError("Invalid permutation") 

 

def _repr_(self): 

""" 

TESTS:: 

 

sage: Permutations(['c','a','c']) 

Permutations of the multi-set ['c', 'a', 'c'] 

""" 

return "Permutations of the multi-set %s"%list(self.mset) 

 

def __iter__(self): 

r""" 

Iterate over ``self``. 

 

EXAMPLES:: 

 

sage: [ p for p in Permutations(['c','t','t'])] # indirect doctest 

[['c', 't', 't'], ['t', 'c', 't'], ['t', 't', 'c']] 

 

TESTS: 

 

The empty multiset:: 

 

sage: list(sage.combinat.permutation.Permutations_mset([])) 

[[]] 

""" 

mset = self.mset 

n = len(mset) 

from array import array 

mset_list = array('I', sorted(mset.index(x) for x in mset)) 

 

yield self.element_class(self, map_to_list(mset_list, mset, n), check=False) 

 

if n <= 1: 

return 

 

while next_perm(mset_list): 

#Yield the permutation 

yield self.element_class(self, map_to_list(mset_list, mset, n), check=False) 

 

def cardinality(self): 

""" 

EXAMPLES:: 

 

sage: Permutations([1,2,2]).cardinality() 

3 

sage: Permutations([1,1,2,2,2]).cardinality() 

10 

""" 

lmset = list(self.mset) 

mset_list = [lmset.index(x) for x in lmset] 

d = {} 

for i in mset_list: 

d[i] = d.get(i, 0) + 1 

 

c = factorial(len(lmset)) 

for i in itervalues(d): 

if i != 1: 

c //= factorial(i) 

return ZZ(c) 

 

class Permutations_set(Permutations): 

""" 

Permutations of an arbitrary given finite set. 

 

Here, a "permutation of a finite set `S`" means a list of the 

elements of `S` in which every element of `S` occurs exactly 

once. This is not to be confused with bijections from `S` to 

`S`, which are also often called permutations in literature. 

""" 

@staticmethod 

def __classcall_private__(cls, s): 

""" 

Normalize arguments to ensure a unique representation. 

 

TESTS:: 

 

sage: S1 = Permutations(['c','a','t']) 

sage: S2 = Permutations(('c','a','t')) 

sage: S1 is S2 

True 

""" 

return super(Permutations_set, cls).__classcall__(cls, tuple(s)) 

 

def __init__(self, s): 

""" 

TESTS:: 

 

sage: S = Permutations(['c','a','t']) 

sage: TestSuite(S).run() 

""" 

Permutations.__init__(self, category=FiniteEnumeratedSets()) 

self._set = s 

 

def __contains__(self, x): 

""" 

EXAMPLES:: 

 

sage: p = Permutations([4,-1,'cthulhu']) 

sage: [4,-1,'cthulhu'] in p 

True 

sage: [] in p 

False 

sage: [4,'cthulhu','fhtagn'] in p 

False 

sage: [4,'cthulhu',4,-1] in p 

False 

sage: [-1,'cthulhu',4] in p 

True 

""" 

s = list(self._set) 

if len(x) != len(s): 

return False 

for i in x: 

if i in s: 

s.remove(i) 

else: 

return False 

return True 

 

def _repr_(self): 

""" 

TESTS:: 

 

sage: Permutations(['c','a','t']) 

Permutations of the set ['c', 'a', 't'] 

""" 

return "Permutations of the set %s"%list(self._set) 

 

class Element(ClonableArray): 

""" 

A permutation of an arbitrary set. 

""" 

def check(self): 

""" 

Verify that ``self`` is a valid permutation of the underlying 

set. 

 

EXAMPLES:: 

 

sage: S = Permutations(['c','a','t']) 

sage: elt = S(['t','c','a']) 

sage: elt.check() 

""" 

if self not in self.parent(): 

raise ValueError("Invalid permutation") 

 

def __iter__(self): 

""" 

Iterate over ``self``. 

 

EXAMPLES:: 

 

sage: S = Permutations(['c','a','t']) 

sage: S.list() 

[['c', 'a', 't'], 

['c', 't', 'a'], 

['a', 'c', 't'], 

['a', 't', 'c'], 

['t', 'c', 'a'], 

['t', 'a', 'c']] 

""" 

for p in itertools.permutations(self._set, len(self._set)): 

yield self.element_class(self, p) 

 

def cardinality(self): 

""" 

EXAMPLES:: 

 

sage: Permutations([1,2,3]).cardinality() 

6 

""" 

return factorial(len(self._set)) 

 

def random_element(self): 

""" 

EXAMPLES:: 

 

sage: Permutations([1,2,3]).random_element() 

[1, 2, 3] 

""" 

return sample(self._set, len(self._set)) 

 

 

class Permutations_msetk(Permutations_mset): 

""" 

Length-`k` partial permutations of a multiset. 

 

A length-`k` partial permutation of a multiset `M` is 

represented by a list of length `k` whose entries are 

elements of `M`, appearing in the list with a multiplicity 

not higher than their respective multiplicity in `M`. 

""" 

@staticmethod 

def __classcall__(cls, mset, k): 

""" 

Normalize arguments to ensure a unique representation. 

 

TESTS:: 

 

sage: S1 = Permutations(['c','a','c'], 2) 

sage: S2 = Permutations(('c','a','c'), 2) 

sage: S1 is S2 

True 

""" 

return super(Permutations_msetk, cls).__classcall__(cls, tuple(mset), k) 

 

def __init__(self, mset, k): 

""" 

TESTS:: 

 

sage: P = Permutations([1,2,2],2) 

sage: TestSuite(P).run() 

""" 

Permutations_mset.__init__(self, mset) 

self.k = k 

 

def __contains__(self, x): 

""" 

EXAMPLES:: 

 

sage: p = Permutations([1,2,2],2) 

sage: [1,2,2] in p 

False 

sage: [2,2] in p 

True 

sage: [1,1] in p 

False 

sage: [2,1] in p 

True 

""" 

if len(x) != self.k: return False 

s = list(self.mset) 

for i in x: 

if i in s: 

s.remove(i) 

else: 

return False 

return True 

 

def _repr_(self): 

""" 

TESTS:: 

 

sage: Permutations([1,2,2],2) 

Permutations of the multi-set [1, 2, 2] of length 2 

""" 

return "Permutations of the multi-set %s of length %s"%(list(self.mset), self.k) 

 

def __iter__(self): 

""" 

EXAMPLES:: 

 

sage: Permutations([1,2,2],2).list() 

[[1, 2], [2, 1], [2, 2]] 

""" 

mset = self.mset 

lmset = list(mset) 

mset_list = [lmset.index(x) for x in lmset] 

indices = eval(gap.eval('Arrangements(%s,%s)'%(mset_list, self.k))) 

for ktuple in indices: 

yield self.element_class(self, [lmset[x] for x in ktuple]) 

 

class Permutations_setk(Permutations_set): 

""" 

Length-`k` partial permutations of an arbitrary given finite set. 

 

Here, a "length-`k` partial permutation of a finite set `S`" means 

a list of length `k` whose entries are pairwise distinct and all 

belong to `S`. 

""" 

@staticmethod 

def __classcall_private__(cls, s, k): 

""" 

Normalize arguments to ensure a unique representation. 

 

TESTS:: 

 

sage: S1 = Permutations(['c','a','t'], 2) 

sage: S2 = Permutations(('c','a','t'), 2) 

sage: S1 is S2 

True 

""" 

return super(Permutations_setk, cls).__classcall__(cls, tuple(s), k) 

 

def __init__(self, s, k): 

""" 

TESTS:: 

 

sage: P = Permutations([1,2,4],2) 

sage: TestSuite(P).run() 

""" 

Permutations_set.__init__(self, s) 

self.k = k 

 

def __contains__(self, x): 

""" 

EXAMPLES:: 

 

sage: p = Permutations([1,2,4],2) 

sage: [1,2,4] in p 

False 

sage: [2,2] in p 

False 

sage: [1,4] in p 

True 

sage: [2,1] in p 

True 

""" 

if len(x) != self.k: 

return False 

s = list(self._set) 

return all(i in s for i in x) and len(uniq(x)) == len(x) 

 

def _repr_(self): 

""" 

TESTS:: 

 

sage: repr(Permutations([1,2,4],2)) 

'Permutations of the set [1, 2, 4] of length 2' 

""" 

return "Permutations of the set %s of length %s"%(list(self._set), self.k) 

 

def __iter__(self): 

""" 

EXAMPLES:: 

 

sage: [i for i in Permutations([1,2,4],2)] 

[[1, 2], [1, 4], [2, 1], [2, 4], [4, 1], [4, 2]] 

""" 

for perm in itertools.permutations(self._set, int(self.k)): 

yield self.element_class(self, perm) 

 

def random_element(self): 

""" 

EXAMPLES:: 

 

sage: Permutations([1,2,4], 2).random_element() 

[1, 2] 

""" 

return sample(self._set, self.k) 

 

################################## 

# Arrangements 

 

class Arrangements(Permutations): 

r""" 

An arrangement of a multiset ``mset`` is an ordered selection 

without repetitions. It is represented by a list that contains 

only elements from ``mset``, but maybe in a different order. 

 

``Arrangements`` returns the combinatorial class of 

arrangements of the multiset ``mset`` that contain ``k`` elements. 

 

EXAMPLES:: 

 

sage: mset = [1,1,2,3,4,4,5] 

sage: Arrangements(mset,2).list() 

[[1, 1], 

[1, 2], 

[1, 3], 

[1, 4], 

[1, 5], 

[2, 1], 

[2, 3], 

[2, 4], 

[2, 5], 

[3, 1], 

[3, 2], 

[3, 4], 

[3, 5], 

[4, 1], 

[4, 2], 

[4, 3], 

[4, 4], 

[4, 5], 

[5, 1], 

[5, 2], 

[5, 3], 

[5, 4]] 

sage: Arrangements(mset,2).cardinality() 

22 

sage: Arrangements( ["c","a","t"], 2 ).list() 

[['c', 'a'], ['c', 't'], ['a', 'c'], ['a', 't'], ['t', 'c'], ['t', 'a']] 

sage: Arrangements( ["c","a","t"], 3 ).list() 

[['c', 'a', 't'], 

['c', 't', 'a'], 

['a', 'c', 't'], 

['a', 't', 'c'], 

['t', 'c', 'a'], 

['t', 'a', 'c']] 

""" 

@staticmethod 

def __classcall_private__(cls, mset, k): 

""" 

Return the correct parent. 

 

EXAMPLES:: 

 

sage: A1 = Arrangements( ["c","a","t"], 2) 

sage: A2 = Arrangements( ("c","a","t"), 2) 

sage: A1 is A2 

True 

""" 

mset = tuple(mset) 

if [mset.index(_) for _ in mset] == list(range(len(mset))): 

return Arrangements_setk(mset, k) 

return Arrangements_msetk(mset, k) 

 

def cardinality(self): 

""" 

Return the cardinality of ``self``. 

 

EXAMPLES:: 

 

sage: A = Arrangements([1,1,2,3,4,4,5], 2) 

sage: A.cardinality() 

22 

""" 

one = ZZ.one() 

return sum(one for p in self) 

 

class Arrangements_msetk(Arrangements, Permutations_msetk): 

r""" 

Arrangements of length `k` of a multiset `M`. 

""" 

def _repr_(self): 

""" 

TESTS:: 

 

sage: Arrangements([1,2,2],2) 

Arrangements of the multi-set [1, 2, 2] of length 2 

""" 

return "Arrangements of the multi-set %s of length %s"%(list(self.mset),self.k) 

 

class Arrangements_setk(Arrangements, Permutations_setk): 

r""" 

Arrangements of length `k` of a set `S`. 

""" 

def _repr_(self): 

""" 

TESTS:: 

 

sage: Arrangements([1,2,3],2) 

Arrangements of the set [1, 2, 3] of length 2 

""" 

return "Arrangements of the set %s of length %s"%(list(self._set),self.k) 

 

############################################################### 

# Standard permutations 

 

class StandardPermutations_all(Permutations): 

""" 

All standard permutations. 

""" 

def __init__(self): 

""" 

TESTS:: 

 

sage: SP = Permutations() 

sage: TestSuite(SP).run() 

""" 

Permutations.__init__(self, category=InfiniteEnumeratedSets()) 

 

def _repr_(self): 

""" 

TESTS:: 

 

sage: Permutations() 

Standard permutations 

""" 

return "Standard permutations" 

 

def __contains__(self, x): 

""" 

TESTS:: 

 

sage: [] in Permutations() 

True 

sage: [1] in Permutations() 

True 

sage: [2] in Permutations() 

False 

sage: [1,2] in Permutations() 

True 

sage: [2,1] in Permutations() 

True 

sage: [1,2,2] in Permutations() 

False 

sage: [3,1,5,2] in Permutations() 

False 

sage: [3,4,1,5,2] in Permutations() 

True 

""" 

if isinstance(x, Permutation): 

return True 

elif isinstance(x, list): 

s = sorted(x[:]) 

if s != list(range(1, len(x)+1)): 

return False 

return True 

else: 

return False 

 

def __iter__(self): 

""" 

Iterate over ``self``. 

 

TESTS:: 

 

sage: it = iter(Permutations()) 

sage: [next(it) for i in range(10)] 

[[], [1], [1, 2], [2, 1], [1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]] 

""" 

n = 0 

while True: 

for p in itertools.permutations(range(1, n + 1)): 

yield self.element_class(self, p) 

n += 1 

 

class StandardPermutations_n_abstract(Permutations): 

""" 

Abstract base class for subsets of permutations of the 

set `\{1, 2, \ldots, n\}`. 

 

.. WARNING:: 

 

Anything inheriting from this class should override the 

``__contains__`` method. 

""" 

def __init__(self, n, category=None): 

""" 

TESTS: 

 

We skip the reduced word method because it does not respect the 

ordering for multiplication:: 

 

sage: SP = Permutations(3) 

sage: TestSuite(SP).run(skip=['_test_reduced_word', 

....: '_test_descents']) 

 

sage: SP.options.mult='r2l' 

sage: TestSuite(SP).run(skip='_test_descents') 

sage: SP.options._reset() 

""" 

self.n = n 

if category is None: 

category = FiniteEnumeratedSets() 

Permutations.__init__(self, category=category) 

 

def _element_constructor_(self, x, check_input=True): 

""" 

Construct an element of ``self`` from ``x``. 

 

TESTS:: 

 

sage: P = Permutations(5) 

sage: P([2,3,1]) 

[2, 3, 1, 4, 5] 

 

sage: G = SymmetricGroup(4) 

sage: P = Permutations(4) 

sage: x = G([4,3,1,2]); x 

(1,4,2,3) 

sage: P(x) 

[4, 3, 1, 2] 

sage: G(P(x)) 

(1,4,2,3) 

 

sage: P = PermutationGroup([[(1,3,5),(2,4)],[(1,3)]]) 

sage: x = P([(3,5),(2,4)]); x 

(2,4)(3,5) 

sage: Permutations(6)(SymmetricGroup(6)(x)) 

[1, 4, 5, 2, 3, 6] 

sage: Permutations(6)(x) # known bug 

[1, 4, 5, 2, 3, 6] 

""" 

if len(x) < self.n: 

x = list(x) + list(range(len(x) + 1, self.n + 1)) 

return self.element_class(self, x, check_input=check_input) 

 

def __contains__(self, x): 

""" 

TESTS:: 

 

sage: [] in Permutations(0) 

True 

sage: [1,2] in Permutations(2) 

True 

sage: [1,2] in Permutations(3) 

False 

sage: [3,2,1] in Permutations(3) 

True 

""" 

return Permutations.__contains__(self, x) and len(x) == self.n 

 

class StandardPermutations_n(StandardPermutations_n_abstract): 

r""" 

Permutations of the set `\{1, 2, \ldots, n\}`. 

 

These are also called permutations of size `n`, or the elements 

of the `n`-th symmetric group. 

 

.. TODO:: 

 

Have a :meth:`reduced_word` which works in both multiplication 

conventions. 

""" 

def __init__(self, n): 

""" 

Initialize ``self``. 

 

TESTS:: 

 

sage: P = Permutations(5) 

sage: P.options.mult = 'r2l' 

sage: TestSuite(P).run(skip='_test_descents') 

sage: P.options._reset() 

""" 

cat = FiniteWeylGroups().Irreducible() & FinitePermutationGroups() 

StandardPermutations_n_abstract.__init__(self, n, category=cat) 

 

def _repr_(self): 

""" 

TESTS:: 

 

sage: Permutations(3) 

Standard permutations of 3 

""" 

return "Standard permutations of %s"%self.n 

 

def __iter__(self): 

""" 

EXAMPLES:: 

 

sage: [p for p in Permutations(0)] 

[[]] 

sage: [p for p in Permutations(3)] 

[[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]] 

""" 

for p in itertools.permutations(range(1, self.n + 1)): 

yield self.element_class(self, p) 

 

def _coerce_map_from_(self, G): 

""" 

Return a coerce map or ``True`` if there exists a coerce map 

from ``G``. 

 

.. WARNING:: 

 

The coerce maps between ``Permutations(n)`` and 

``SymmetricGroup(n)`` exist, but do not respect the 

multiplication when the global variable 

``Permutations.options.mult`` (see 

:meth:`sage.combinat.permutation.Permutations.options` ) 

is set to ``'r2l'``. (Indeed, the multiplication 

in ``Permutations(n)`` depends on this global 

variable, while the multiplication in 

``SymmetricGroup(n)`` does not.) 

 

EXAMPLES:: 

 

sage: P = Permutations(6) 

sage: P.has_coerce_map_from(SymmetricGroup(6)) 

True 

sage: P.has_coerce_map_from(SymmetricGroup(5)) 

True 

sage: P.has_coerce_map_from(SymmetricGroup(7)) 

False 

sage: P.has_coerce_map_from(Permutations(5)) 

True 

sage: P.has_coerce_map_from(Permutations(7)) 

False 

""" 

if isinstance(G, SymmetricGroup): 

D = G.domain() 

if len(D) > self.n or list(D) != list(range(1, len(D)+1)): 

return False 

return self._from_permutation_group_element 

if isinstance(G, StandardPermutations_n) and G.n <= self.n: 

return True 

return super(StandardPermutations_n, self)._coerce_map_from_(G) 

 

def _from_permutation_group_element(self, x): 

""" 

Return an element of ``self`` from a permutation group element. 

 

TESTS:: 

 

sage: P = Permutations(4) 

sage: G = SymmetricGroup(4) 

sage: x = G([4,3,1,2]) 

sage: P._from_permutation_group_element(x) 

[4, 3, 1, 2] 

""" 

return self(x.domain()) 

 

def identity(self): 

r""" 

Return the identity permutation of size `n`. 

 

EXAMPLES:: 

 

sage: Permutations(4).identity() 

[1, 2, 3, 4] 

sage: Permutations(0).identity() 

[] 

""" 

return self.element_class(self, range(1,self.n+1)) 

 

one = identity 

 

def unrank(self, r): 

""" 

EXAMPLES:: 

 

sage: SP3 = Permutations(3) 

sage: l = list(map(SP3.unrank, range(6))) 

sage: l == SP3.list() 

True 

sage: SP0 = Permutations(0) 

sage: l = list(map(SP0.unrank, range(1))) 

sage: l == SP0.list() 

True 

""" 

if r >= factorial(self.n) or r < 0: 

raise ValueError 

else: 

return from_rank(self.n, r) 

 

def rank(self, p=None): 

""" 

Return the rank of ``self`` or ``p`` depending on input. 

 

If a permutation ``p`` is given, return the rank of ``p`` 

in ``self``. Otherwise return the dimension of the 

underlying vector space spanned by the (simple) roots. 

 

EXAMPLES:: 

 

sage: P = Permutations(5) 

sage: P.rank() 

4 

 

sage: SP3 = Permutations(3) 

sage: list(map(SP3.rank, SP3)) 

[0, 1, 2, 3, 4, 5] 

sage: SP0 = Permutations(0) 

sage: list(map(SP0.rank, SP0)) 

[0] 

""" 

if p is None: 

return self.n - 1 

if p in self: 

return Permutation(p).rank() 

raise ValueError("x not in self") 

 

def random_element(self): 

""" 

EXAMPLES:: 

 

sage: Permutations(4).random_element() 

[1, 2, 4, 3] 

""" 

return self.element_class(self, sample(range(1,self.n+1), self.n)) 

 

def cardinality(self): 

""" 

Return the number of permutations of size `n`, which is `n!`. 

 

EXAMPLES:: 

 

sage: Permutations(0).cardinality() 

1 

sage: Permutations(3).cardinality() 

6 

sage: Permutations(4).cardinality() 

24 

""" 

return factorial(self.n) 

 

def degree(self): 

""" 

Return the degree of ``self``. 

 

This is the cardinality `n` of the set ``self`` acts on. 

 

EXAMPLES:: 

 

sage: Permutations(0).degree() 

0 

sage: Permutations(1).degree() 

1 

sage: Permutations(5).degree() 

5 

""" 

return self.n 

 

def degrees(self): 

""" 

Return the degrees of ``self``. 

 

These are the degrees of the fundamental invariants of the 

ring of polynomial invariants. 

 

EXAMPLES:: 

 

sage: Permutations(3).degrees() 

(2, 3) 

sage: Permutations(7).degrees() 

(2, 3, 4, 5, 6, 7) 

""" 

return tuple(Integer(i) for i in range(2, self.n+1)) 

 

def codegrees(self): 

""" 

Return the codegrees of ``self``. 

 

EXAMPLES:: 

 

sage: Permutations(3).codegrees() 

(0, 1) 

sage: Permutations(7).codegrees() 

(0, 1, 2, 3, 4, 5) 

""" 

return tuple(Integer(i) for i in range(self.n-1)) 

 

def element_in_conjugacy_classes(self, nu): 

r""" 

Return a permutation with cycle type ``nu``. 

 

If the size of ``nu`` is smaller than the size of permutations in 

``self``, then some fixed points are added. 

 

EXAMPLES:: 

 

sage: PP = Permutations(5) 

sage: PP.element_in_conjugacy_classes([2,2]) 

[2, 1, 4, 3, 5] 

""" 

from sage.combinat.partition import Partition 

nu = Partition(nu) 

if nu.size() > self.n: 

raise ValueError("The size of the partition (=%s) should be lower or equal" 

" to the size of the permutations (=%s)"%(nu.size,self.n)) 

l = [] 

i = 0 

for nui in nu: 

for j in range(nui-1): 

l.append(i+j+2) 

l.append(i+1) 

i += nui 

for i in range(nu.size(), self.n): 

l.append(i+1) 

return self.element_class(self, l) 

 

def conjugacy_classes_representatives(self): 

""" 

Return a complete list of representatives of conjugacy classes 

in ``self``. 

 

Let `S_n` be the symmetric group on `n` letters. The conjugacy 

classes are indexed by partitions `\lambda` of `n`. The ordering 

of the conjugacy classes is reverse lexicographic order of 

the partitions. 

 

EXAMPLES:: 

 

sage: G = Permutations(5) 

sage: G.conjugacy_classes_representatives() 

[[1, 2, 3, 4, 5], 

[2, 1, 3, 4, 5], 

[2, 1, 4, 3, 5], 

[2, 3, 1, 4, 5], 

[2, 3, 1, 5, 4], 

[2, 3, 4, 1, 5], 

[2, 3, 4, 5, 1]] 

 

TESTS: 

 

Check some border cases:: 

 

sage: S = Permutations(0) 

sage: S.conjugacy_classes_representatives() 

[[]] 

sage: S = Permutations(1) 

sage: S.conjugacy_classes_representatives() 

[[1]] 

""" 

from sage.combinat.partition import Partitions_n 

return [ self.element_in_conjugacy_classes(la) 

for la in reversed(Partitions_n(self.n)) ] 

 

def conjugacy_classes_iterator(self): 

""" 

Iterate over the conjugacy classes of ``self``. 

 

EXAMPLES:: 

 

sage: G = Permutations(4) 

sage: list(G.conjugacy_classes_iterator()) == G.conjugacy_classes() 

True 

""" 

from sage.combinat.partition import Partitions_n 

from sage.groups.perm_gps.symgp_conjugacy_class import PermutationsConjugacyClass 

for la in reversed(Partitions_n(self.n)): 

yield PermutationsConjugacyClass(self, la) 

 

def conjugacy_classes(self): 

""" 

Return a list of the conjugacy classes of ``self``. 

 

EXAMPLES:: 

 

sage: G = Permutations(4) 

sage: G.conjugacy_classes() 

[Conjugacy class of cycle type [1, 1, 1, 1] in Standard permutations of 4, 

Conjugacy class of cycle type [2, 1, 1] in Standard permutations of 4, 

Conjugacy class of cycle type [2, 2] in Standard permutations of 4, 

Conjugacy class of cycle type [3, 1] in Standard permutations of 4, 

Conjugacy class of cycle type [4] in Standard permutations of 4] 

""" 

return list(self.conjugacy_classes_iterator()) 

 

def conjugacy_class(self, g): 

r""" 

Return the conjugacy class of ``g`` in ``self``. 

 

INPUT: 

 

- ``g`` -- a partition or an element of ``self`` 

 

EXAMPLES:: 

 

sage: G = Permutations(5) 

sage: g = G([2,3,4,1,5]) 

sage: G.conjugacy_class(g) 

Conjugacy class of cycle type [4, 1] in Standard permutations of 5 

sage: G.conjugacy_class(Partition([2, 1, 1, 1])) 

Conjugacy class of cycle type [2, 1, 1, 1] in Standard permutations of 5 

""" 

from sage.groups.perm_gps.symgp_conjugacy_class import PermutationsConjugacyClass 

return PermutationsConjugacyClass(self, g) 

 

def algebra(self, base_ring, category=None): 

""" 

Return the symmetric group algebra associated to ``self``. 

 

INPUT: 

 

- ``base_ring`` -- a ring 

- ``category`` -- a category (default: the category of ``self``) 

 

EXAMPLES:: 

 

sage: P = Permutations(4) 

sage: A = P.algebra(QQ); A 

Symmetric group algebra of order 4 over Rational Field 

 

sage: A.category() 

Join of Category of coxeter group algebras over Rational Field 

and Category of finite group algebras over Rational Field 

sage: A = P.algebra(QQ, category=Monoids()) 

sage: A.category() 

Category of finite dimensional monoid algebras over Rational Field 

""" 

from sage.combinat.symmetric_group_algebra import SymmetricGroupAlgebra 

return SymmetricGroupAlgebra(base_ring, self, category=category) 

 

@cached_method 

def index_set(self): 

""" 

Return the index set for the descents of the symmetric group ``self``. 

 

This is `\{ 1, 2, \ldots, n-1 \}`, where ``self`` is `S_n`. 

 

EXAMPLES:: 

 

sage: P = Permutations(8) 

sage: P.index_set() 

(1, 2, 3, 4, 5, 6, 7) 

""" 

return tuple(range(1, self.n)) 

 

def cartan_type(self): 

r""" 

Return the Cartan type of ``self``. 

 

The symmetric group `S_n` is a Coxeter group of type `A_{n-1}`. 

 

EXAMPLES:: 

 

sage: A = SymmetricGroup([2,3,7]); A.cartan_type() 

['A', 2] 

sage: A = SymmetricGroup([]); A.cartan_type() 

['A', 0] 

""" 

from sage.combinat.root_system.cartan_type import CartanType 

return CartanType(['A', max(self.n - 1,0)]) 

 

def simple_reflection(self, i): 

r""" 

For `i` in the index set of ``self`` (that is, for `i` in 

`\{ 1, 2, \ldots, n-1 \}`, where ``self`` is `S_n`), this 

returns the elementary transposition `s_i = (i,i+1)`. 

 

EXAMPLES:: 

 

sage: P = Permutations(4) 

sage: P.simple_reflection(2) 

[1, 3, 2, 4] 

sage: P.simple_reflections() 

Finite family {1: [2, 1, 3, 4], 2: [1, 3, 2, 4], 3: [1, 2, 4, 3]} 

""" 

g = list(range(1, self.n+1)) 

g[i-1] = i+1 

g[i] = i 

return self.element_class(self, g) 

 

class Element(Permutation): 

def has_left_descent(self, i, mult=None): 

r""" 

Check if ``i`` is a left descent of ``self``. 

 

A *left descent* of a permutation `\pi \in S_n` means an index 

`i \in \{ 1, 2, \ldots, n-1 \}` such that 

`s_i \circ \pi` has smaller length than `\pi`. Here, `\circ` 

denotes the multiplication of `S_n`. How it is defined depends 

on the ``mult`` variable in 

:meth:`Permutations.options`. If this variable is set 

to ``'l2r'``, then the multiplication is defined by the rule 

`(\alpha \beta) (x) = \beta( \alpha (x) )` for `\alpha, 

\beta \in S_n` and `x \in \{ 1, 2, \ldots, n \}`; then, a left 

descent of `\pi` is an index `i \in \{ 1, 2, \ldots, n-1 \}` 

satisfying `\pi(i) > \pi(i+1)`. If this variable is set 

to ``'r2l'``, then the multiplication is defined by the rule 

`(\alpha \beta) (x) = \alpha( \beta (x) )` for `\alpha, 

\beta \in S_n` and `x \in \{ 1, 2, \ldots, n \}`; then, a left 

descent of `\pi` is an index `i \in \{ 1, 2, \ldots, n-1 \}` 

satisfying `\pi^{-1}(i) > \pi^{-1}(i+1)`. 

 

The optional parameter ``mult`` can be set to ``'l2r'`` or 

``'r2l'``; if so done, it is used instead of the ``mult`` 

variable in :meth:`Permutations.options`. Anyone using 

this method in a non-interactive environment is encouraged to 

do so in order to have code behave reliably. 

 

.. WARNING:: 

 

The methods :meth:`descents` and :meth:`idescents` behave 

differently than their Weyl group counterparts. In 

particular, the indexing is 0-based. This could lead to 

errors. Instead, construct the descent set as in the example. 

 

.. WARNING:: 

 

The optional input ``mult`` might disappear once :trac:`14881` 

is fixed. 

 

EXAMPLES:: 

 

sage: P = Permutations(4) 

sage: x = P([3, 2, 4, 1]) 

sage: x.descents() 

[1, 3] 

sage: [i for i in P.index_set() if x.has_left_descent(i)] 

[1, 3] 

sage: [i for i in P.index_set() if x.has_left_descent(i, mult="l2r")] 

[1, 3] 

sage: [i for i in P.index_set() if x.has_left_descent(i, mult="r2l")] 

[1, 2] 

""" 

if mult is None: 

mult = self.parent().options.mult 

if mult != 'l2r': 

self = self.inverse() 

return self[i-1] > self[i] 

 

def has_right_descent(self, i, mult=None): 

r""" 

Check if ``i`` is a right descent of ``self``. 

 

A *right descent* of a permutation `\pi \in S_n` means an index 

`i \in \{ 1, 2, \ldots, n-1 \}` such that 

`\pi \circ s_i` has smaller length than `\pi`. Here, `\circ` 

denotes the multiplication of `S_n`. How it is defined depends 

on the ``mult`` variable in 

:meth:`Permutations.options`. If this variable is set 

to ``'l2r'``, then the multiplication is defined by the rule 

`(\alpha \beta) (x) = \beta( \alpha (x) )` for `\alpha, 

\beta \in S_n` and `x \in \{ 1, 2, \ldots, n \}`; then, a right 

descent of `\pi` is an index `i \in \{ 1, 2, \ldots, n-1 \}` 

satisfying `\pi^{-1}(i) > \pi^{-1}(i+1)`. If this variable is 

set to ``'r2l'``, then the multiplication is defined by the 

rule `(\alpha \beta) (x) = \alpha( \beta (x) )` for `\alpha, 

\beta \in S_n` and `x \in \{ 1, 2, \ldots, n \}`; then, a right 

descent of `\pi` is an index `i \in \{ 1, 2, \ldots, n-1 \}` 

satisfying `\pi(i) > \pi(i+1)`. 

 

The optional parameter ``mult`` can be set to ``'l2r'`` or 

``'r2l'``; if so done, it is used instead of the ``mult`` 

variable in :meth:`Permutations.options`. Anyone using 

this method in a non-interactive environment is encouraged to 

do so in order to have code behave reliably. 

 

.. WARNING:: 

 

The methods :meth:`descents` and :meth:`idescents` behave 

differently than their Weyl group counterparts. In 

particular, the indexing is 0-based. This could lead to 

errors. Instead, construct the descent set as in the example. 

 

.. WARNING:: 

 

The optional input ``mult`` might disappear once :trac:`14881` 

is fixed. 

 

EXAMPLES:: 

 

sage: P = Permutations(4) 

sage: x = P([3, 2, 4, 1]) 

sage: (~x).descents() 

[1, 2] 

sage: [i for i in P.index_set() if x.has_right_descent(i)] 

[1, 2] 

sage: [i for i in P.index_set() if x.has_right_descent(i, mult="l2r")] 

[1, 2] 

sage: [i for i in P.index_set() if x.has_right_descent(i, mult="r2l")] 

[1, 3] 

""" 

if mult is None: 

mult = self.parent().options.mult 

if mult != 'r2l': 

self = self.inverse() 

return self[i-1] > self[i] 

 

def __mul__(self, other): 

r""" 

Multiply ``self`` and ``other``. 

 

EXAMPLES:: 

 

sage: P = Permutations(4) 

sage: P.simple_reflection(1) * Permutation([6,5,4,3,2,1]) 

[5, 6, 4, 3, 2, 1] 

sage: Permutations.options.mult='r2l' 

sage: P.simple_reflection(1) * Permutation([6,5,4,3,2,1]) 

[6, 5, 4, 3, 1, 2] 

sage: Permutations.options.mult='l2r' 

""" 

if not isinstance(other, StandardPermutations_n.Element): 

return Permutation.__mul__(self, other) 

if other.parent() is not self.parent(): 

# They have different parents (but both are (like) Permutations of n) 

mul_order = self.parent().options.mult 

if mul_order == 'l2r': 

p = right_action_product(self._list, other._list) 

elif mul_order == 'r2l': 

p = left_action_product(self._list, other._list) 

return Permutations(len(p))(p) 

# They have the same parent 

return self._mul_(other) 

 

def _mul_(self, other): 

r""" 

Multiply ``self`` and ``other``. 

 

EXAMPLES:: 

 

sage: P = Permutations(4) 

sage: P.prod(P.group_generators()).parent() is P 

True 

""" 

mul_order = self.parent().options.mult 

if mul_order == 'l2r': 

p = right_action_same_n(self._list, other._list) 

elif mul_order == 'r2l': 

p = left_action_same_n(self._list, other._list) 

return self.__class__(self.parent(), p) 

 

@combinatorial_map(order=2, name='inverse') 

def inverse(self): 

r""" 

Return the inverse of ``self``. 

 

EXAMPLES:: 

 

sage: P = Permutations(4) 

sage: w0 = P([4,3,2,1]) 

sage: w0.inverse() == w0 

True 

sage: w0.inverse().parent() is P 

True 

sage: P([3,2,4,1]).inverse() 

[4, 2, 1, 3] 

""" 

w = list(range(len(self))) 

for i,j in enumerate(self): 

w[j-1] = i+1 

return self.__class__(self.parent(), w) 

 

__invert__ = inverse 

 

############################# 

# Constructing Permutations # 

############################# 

 

# TODO: Make this a coercion 

def from_permutation_group_element(pge, parent=None): 

""" 

Return a :class:`Permutation` given a :class:`PermutationGroupElement` 

``pge``. 

 

EXAMPLES:: 

 

sage: import sage.combinat.permutation as permutation 

sage: pge = PermutationGroupElement([(1,2),(3,4)]) 

sage: permutation.from_permutation_group_element(pge) 

[2, 1, 4, 3] 

""" 

if not isinstance(pge, PermutationGroupElement): 

raise TypeError("pge (= %s) must be a PermutationGroupElement"%pge) 

 

if parent is None: 

parent = Permutations( len(pge.domain()) ) 

 

return parent(pge.domain()) 

 

def from_rank(n, rank): 

r""" 

Return the permutation of the set `\{1,...,n\}` with lexicographic 

rank ``rank``. This is the permutation whose Lehmer code is the 

factoradic representation of ``rank``. In particular, the 

permutation with rank `0` is the identity permutation. 

 

The permutation is computed without iterating through all of the 

permutations with lower rank. This makes it efficient for large 

permutations. 

 

.. NOTE:: 

 

The variable ``rank`` is not checked for being in the interval 

from `0` to `n! - 1`. When outside this interval, it acts as 

its residue modulo `n!`. 

 

EXAMPLES:: 

 

sage: import sage.combinat.permutation as permutation 

sage: Permutation([3, 6, 5, 4, 2, 1]).rank() 

359 

sage: [permutation.from_rank(3, i) for i in range(6)] 

[[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]] 

sage: Permutations(6)[10] 

[1, 2, 4, 6, 3, 5] 

sage: permutation.from_rank(6,10) 

[1, 2, 4, 6, 3, 5] 

""" 

#Find the factoradic of rank 

factoradic = [None] * n 

for j in range(1,n+1): 

factoradic[n-j] = Integer(rank % j) 

rank = int(rank) // j 

 

return from_lehmer_code(factoradic, Permutations(n)) 

 

def from_inversion_vector(iv, parent=None): 

r""" 

Return the permutation corresponding to inversion vector ``iv``. 

 

See `~sage.combinat.permutation.Permutation.to_inversion_vector` 

for a definition of the inversion vector of a permutation. 

 

EXAMPLES:: 

 

sage: import sage.combinat.permutation as permutation 

sage: permutation.from_inversion_vector([3,1,0,0,0]) 

[3, 2, 4, 1, 5] 

sage: permutation.from_inversion_vector([2,3,6,4,0,2,2,1,0]) 

[5, 9, 1, 8, 2, 6, 4, 7, 3] 

sage: permutation.from_inversion_vector([0]) 

[1] 

sage: permutation.from_inversion_vector([]) 

[] 

""" 

p = iv[:] 

open_spots = list(range(len(iv))) 

for i,ivi in enumerate(iv): 

p[open_spots.pop(ivi)] = i+1 

 

if parent is None: 

parent = Permutations() 

return parent(p) 

 

def from_cycles(n, cycles, parent=None): 

r""" 

Return the permutation in the `n`-th symmetric group whose decomposition 

into disjoint cycles is ``cycles``. 

 

This function checks that its input is correct (i.e. that the cycles are 

disjoint and their elements integers among `1...n`). It raises an exception 

otherwise. 

 

.. WARNING:: 

 

It assumes that the elements are of ``int`` type. 

 

EXAMPLES:: 

 

sage: import sage.combinat.permutation as permutation 

sage: permutation.from_cycles(4, [[1,2]]) 

[2, 1, 3, 4] 

sage: permutation.from_cycles(4, [[1,2,4]]) 

[2, 4, 3, 1] 

sage: permutation.from_cycles(10, [[3,1],[4,5],[6,8,9]]) 

[3, 2, 1, 5, 4, 8, 7, 9, 6, 10] 

sage: permutation.from_cycles(10, ((2, 5), (6, 1, 3))) 

[3, 5, 6, 4, 2, 1, 7, 8, 9, 10] 

sage: permutation.from_cycles(4, []) 

[1, 2, 3, 4] 

sage: permutation.from_cycles(4, [[]]) 

[1, 2, 3, 4] 

sage: permutation.from_cycles(0, []) 

[] 

 

Bad input (see :trac:`13742`):: 

 

sage: Permutation("(-12,2)(3,4)") 

Traceback (most recent call last): 

... 

ValueError: All elements should be strictly positive integers, and I just found a non-positive one. 

sage: Permutation("(1,2)(2,4)") 

Traceback (most recent call last): 

... 

ValueError: An element appears twice. It should not. 

sage: permutation.from_cycles(4, [[1,18]]) 

Traceback (most recent call last): 

... 

ValueError: You claimed that this was a permutation on 1...4 but it contains 18 

""" 

if parent is None: 

parent = Permutations(n) 

 

p = list(range(1, n+1)) 

 

# Is it really a permutation on 1...n ? 

flattened_and_sorted = [] 

for c in cycles: 

flattened_and_sorted.extend(c) 

flattened_and_sorted.sort() 

 

# Empty input 

if not flattened_and_sorted: 

return parent(p, check_input=False) 

 

# Only positive elements 

if int(flattened_and_sorted[0]) < 1: 

raise ValueError("All elements should be strictly positive " 

"integers, and I just found a non-positive one.") 

 

# Really smaller or equal to n ? 

if flattened_and_sorted[-1] > n: 

raise ValueError("You claimed that this was a permutation on 1..."+ 

str(n)+" but it contains "+str(flattened_and_sorted[-1])) 

 

# Disjoint cycles ? 

previous = flattened_and_sorted[0] - 1 

for i in flattened_and_sorted: 

if i == previous: 

raise ValueError("An element appears twice. It should not.") 

else: 

previous = i 

 

for cycle in cycles: 

if not cycle: 

continue 

first = cycle[0] 

for i in range(len(cycle)-1): 

p[cycle[i]-1] = cycle[i+1] 

p[cycle[-1]-1] = first 

 

return parent(p, check_input=False) 

 

def from_lehmer_code(lehmer, parent=None): 

r""" 

Return the permutation with Lehmer code ``lehmer``. 

 

EXAMPLES:: 

 

sage: import sage.combinat.permutation as permutation 

sage: lc = Permutation([2,1,5,4,3]).to_lehmer_code(); lc 

[1, 0, 2, 1, 0] 

sage: permutation.from_lehmer_code(lc) 

[2, 1, 5, 4, 3] 

""" 

p = [] 

open_spots = list(range(1,len(lehmer)+1)) 

for ivi in lehmer: 

p.append(open_spots.pop(ivi)) 

 

if parent is None: 

parent = Permutations() 

return parent(p) 

 

def from_reduced_word(rw, parent=None): 

r""" 

Return the permutation corresponding to the reduced word ``rw``. 

 

See 

:meth:`~sage.combinat.permutation.Permutation.reduced_words` for 

a definition of reduced words and the convention on the order of 

multiplication used. 

 

EXAMPLES:: 

 

sage: import sage.combinat.permutation as permutation 

sage: permutation.from_reduced_word([3,2,3,1,2,3,1]) 

[3, 4, 2, 1] 

sage: permutation.from_reduced_word([]) 

[] 

""" 

if parent is None: 

parent = Permutations() 

 

if not rw: 

return parent([]) 

 

p = [i+1 for i in range(max(rw)+1)] 

 

for i in rw: 

(p[i-1], p[i]) = (p[i], p[i-1]) 

 

return parent(p) 

 

def bistochastic_as_sum_of_permutations(M, check = True): 

r""" 

Return the positive sum of permutations corresponding to 

the bistochastic matrix ``M``. 

 

A stochastic matrix is a matrix with nonnegative real entries such that the 

sum of the elements of any row is equal to `1`. A bistochastic matrix is a 

stochastic matrix whose transpose matrix is also stochastic ( there are 

conditions both on the rows and on the columns ). 

 

According to the Birkhoff-von Neumann Theorem, any bistochastic matrix 

can be written as a convex combination of permutation matrices, which also 

means that the polytope of bistochastic matrices is integer. 

 

As a non-bistochastic matrix can obviously not be written as a convex 

combination of permutations, this theorem is an equivalence. 

 

This function, given a bistochastic matrix, returns the corresponding 

decomposition. 

 

INPUT: 

 

- ``M`` -- A bistochastic matrix 

 

- ``check`` (boolean) -- set to ``True`` (default) to check 

that the matrix is indeed bistochastic 

 

OUTPUT: 

 

- An element of ``CombinatorialFreeModule``, which is a free `F`-module 

( where `F` is the ground ring of the given matrix ) whose basis is 

indexed by the permutations. 

 

.. NOTE:: 

 

- In this function, we just assume 1 to be any constant : for us a 

matrix `M` is bistochastic if there exists `c>0` such that `M/c` 

is bistochastic. 

 

- You can obtain a sequence of pairs ``(permutation,coeff)``, where 

``permutation`` is a Sage ``Permutation`` instance, and ``coeff`` 

its corresponding coefficient from the result of this function 

by applying the ``list`` function. 

 

- If you are interested in the matrix corresponding to a ``Permutation`` 

you will be glad to learn about the ``Permutation.to_matrix()`` method. 

 

- The base ring of the matrix can be anything that can be coerced to ``RR``. 

 

.. SEEALSO:: 

 

- :meth:`~sage.matrix.matrix2.as_sum_of_permutations` 

to use this method through the ``Matrix`` class. 

 

EXAMPLES: 

 

We create a bistochastic matrix from a convex sum of permutations, then 

try to deduce the decomposition from the matrix:: 

 

sage: from sage.combinat.permutation import bistochastic_as_sum_of_permutations 

sage: L = [] 

sage: L.append((9,Permutation([4, 1, 3, 5, 2]))) 

sage: L.append((6,Permutation([5, 3, 4, 1, 2]))) 

sage: L.append((3,Permutation([3, 1, 4, 2, 5]))) 

sage: L.append((2,Permutation([1, 4, 2, 3, 5]))) 

sage: M = sum([c * p.to_matrix() for (c,p) in L]) 

sage: decomp = bistochastic_as_sum_of_permutations(M) 

sage: print(decomp) 

2*B[[1, 4, 2, 3, 5]] + 3*B[[3, 1, 4, 2, 5]] + 9*B[[4, 1, 3, 5, 2]] + 6*B[[5, 3, 4, 1, 2]] 

 

An exception is raised when the matrix is not positive and bistochastic:: 

 

sage: M = Matrix([[2,3],[2,2]]) 

sage: decomp = bistochastic_as_sum_of_permutations(M) 

Traceback (most recent call last): 

... 

ValueError: The matrix is not bistochastic 

 

sage: bistochastic_as_sum_of_permutations(Matrix(GF(7), 2, [2,1,1,2])) 

Traceback (most recent call last): 

... 

ValueError: The base ring of the matrix must have a coercion map to RR 

 

sage: bistochastic_as_sum_of_permutations(Matrix(ZZ, 2, [2,-1,-1,2])) 

Traceback (most recent call last): 

... 

ValueError: The matrix should have nonnegative entries 

""" 

from sage.graphs.bipartite_graph import BipartiteGraph 

from sage.combinat.free_module import CombinatorialFreeModule 

from sage.rings.all import RR 

 

n = M.nrows() 

 

if n != M.ncols(): 

raise ValueError("The matrix is expected to be square") 

 

if not all(x >= 0 for x in M.list()): 

raise ValueError("The matrix should have nonnegative entries") 

 

if check and not M.is_bistochastic(normalized = False): 

raise ValueError("The matrix is not bistochastic") 

 

if not RR.has_coerce_map_from(M.base_ring()): 

raise ValueError("The base ring of the matrix must have a coercion map to RR") 

 

CFM = CombinatorialFreeModule(M.base_ring(), Permutations(n)) 

value = 0 

 

G = BipartiteGraph(M, weighted=True) 

P = Permutations() 

 

while G.size() > 0: 

matching = G.matching(use_edge_labels=True) 

 

# This minimum is strictly larger than 0 

minimum = min([x[2] for x in matching]) 

 

for (u,v,l) in matching: 

if minimum == l: 

G.delete_edge((u,v,l)) 

else: 

G.set_edge_label(u,v,l-minimum) 

 

matching.sort(key=lambda x: x[0]) 

value += minimum * CFM(P([x[1]-n+1 for x in matching])) 

 

return value 

 

 

def bounded_affine_permutation(A): 

r""" 

Return the bounded affine permutation of a matrix. 

 

The *bounded affine permutation* of a matrix `A` with entries in `R` 

is a partial permutation of length `n`, where `n` is the number of 

columns of `A`. The entry in position `i` is the smallest value `j` 

such that column `i` is in the span of columns `i+1, \ldots, j`, 

over `R`, where column indices are taken modulo `n`. 

If column `i` is the zero vector, then the permutation has a 

fixed point at `i`. 

 

INPUT: 

 

- ``A`` -- matrix with entries in a ring `R` 

 

EXAMPLES:: 

 

sage: from sage.combinat.permutation import bounded_affine_permutation 

sage: A = Matrix(ZZ, [[1,0,0,0], [0,1,0,0]]) 

sage: bounded_affine_permutation(A) 

[5, 6, 3, 4] 

 

sage: A = Matrix(ZZ, [[0,1,0,1,0], [0,0,1,1,0]]) 

sage: bounded_affine_permutation(A) 

[1, 4, 7, 8, 5] 

 

REFERENCES: 

 

- [KLS2013]_ 

""" 

n = A.ncols() 

R = A.base_ring() 

from sage.modules.free_module import FreeModule 

from sage.modules.free_module import span 

z = FreeModule(R, A.nrows()).zero() 

v = A.columns() 

perm = [] 

for j in range(n): 

if not v[j]: 

perm.append(j + 1) 

continue 

V = span([z], R) 

for i in range(j + 1, j + n + 1): 

index = i % n 

V = V + span([v[index]], R) 

if not V.dimension(): 

continue 

if v[j] in V: 

perm.append(i + 1) 

break 

S = Permutations(2 * n, n) 

return S(perm) 

 

 

class StandardPermutations_descents(StandardPermutations_n_abstract): 

""" 

Permutations of `\{1, \ldots, n\}` with a fixed set of descents. 

""" 

@staticmethod 

def __classcall_private__(cls, d, n): 

""" 

Normalize input to ensure a unique representation. 

 

EXAMPLES:: 

 

sage: P1 = Permutations(descents=([1,0,4,8],12)) 

sage: P2 = Permutations(descents=((1,0,4,8),12)) 

sage: P1 is P2 

True 

""" 

return super(StandardPermutations_descents, cls).__classcall__(cls, tuple(d), n) 

 

def __init__(self, d, n): 

""" 

The class of all permutations of `\{1, 2, ..., n\}` 

with set of descent positions `d` (where the descent positions 

are being counted from `0`, so that `i` lies in this set if 

and only if the permutation takes a larger value at `i + 1` than 

at `i + 2`). 

 

TESTS:: 

 

sage: P = Permutations(descents=([1,0,2], 5)) 

sage: TestSuite(P).run() 

""" 

StandardPermutations_n_abstract.__init__(self, n) 

self.d = d 

 

def _repr_(self): 

""" 

TESTS:: 

 

sage: Permutations(descents=([1,0,4,8],12)) 

Standard permutations of 12 with descents [1, 0, 4, 8] 

""" 

return "Standard permutations of %s with descents %s"%(self.n, list(self.d)) 

 

def cardinality(self): 

""" 

Return the cardinality of ``self``. 

 

EXAMPLES:: 

 

sage: P = Permutations(descents=([1,0,2],5)) 

sage: P.cardinality() 

4 

""" 

one = ZZ.one() 

return sum(one for p in self) 

 

def first(self): 

""" 

Return the first permutation with descents `d`. 

 

EXAMPLES:: 

 

sage: Permutations(descents=([1,0,4,8],12)).first() 

[3, 2, 1, 4, 6, 5, 7, 8, 10, 9, 11, 12] 

""" 

return descents_composition_first(Composition(descents=(self.d,self.n))) 

 

def last(self): 

""" 

Return the last permutation with descents `d`. 

 

EXAMPLES:: 

 

sage: Permutations(descents=([1,0,4,8],12)).last() 

[12, 11, 8, 9, 10, 4, 5, 6, 7, 1, 2, 3] 

""" 

return descents_composition_last(Composition(descents=(self.d,self.n))) 

 

def __iter__(self): 

""" 

Iterate over all the permutations that have the descents `d`. 

 

EXAMPLES:: 

 

sage: Permutations(descents=([2,0],5)).list() 

[[2, 1, 4, 3, 5], 

[2, 1, 5, 3, 4], 

[3, 1, 4, 2, 5], 

[3, 1, 5, 2, 4], 

[4, 1, 3, 2, 5], 

[5, 1, 3, 2, 4], 

[4, 1, 5, 2, 3], 

[5, 1, 4, 2, 3], 

[3, 2, 4, 1, 5], 

[3, 2, 5, 1, 4], 

[4, 2, 3, 1, 5], 

[5, 2, 3, 1, 4], 

[4, 2, 5, 1, 3], 

[5, 2, 4, 1, 3], 

[4, 3, 5, 1, 2], 

[5, 3, 4, 1, 2]] 

""" 

return iter( descents_composition_list(Composition(descents=(self.d,self.n))) ) 

 

def descents_composition_list(dc): 

""" 

Return a list of all the permutations that have the descent 

composition ``dc``. 

 

EXAMPLES:: 

 

sage: import sage.combinat.permutation as permutation 

sage: permutation.descents_composition_list([1,2,2]) 

[[2, 1, 4, 3, 5], 

[2, 1, 5, 3, 4], 

[3, 1, 4, 2, 5], 

[3, 1, 5, 2, 4], 

[4, 1, 3, 2, 5], 

[5, 1, 3, 2, 4], 

[4, 1, 5, 2, 3], 

[5, 1, 4, 2, 3], 

[3, 2, 4, 1, 5], 

[3, 2, 5, 1, 4], 

[4, 2, 3, 1, 5], 

[5, 2, 3, 1, 4], 

[4, 2, 5, 1, 3], 

[5, 2, 4, 1, 3], 

[4, 3, 5, 1, 2], 

[5, 3, 4, 1, 2]] 

""" 

return [p.inverse() for p in StandardPermutations_recoils(dc)] 

 

def descents_composition_first(dc): 

r""" 

Compute the smallest element of a descent class having a descent 

composition ``dc``. 

 

EXAMPLES:: 

 

sage: import sage.combinat.permutation as permutation 

sage: permutation.descents_composition_first([1,1,3,4,3]) 

[3, 2, 1, 4, 6, 5, 7, 8, 10, 9, 11, 12] 

""" 

if not isinstance(dc, Composition): 

try: 

dc = Composition(dc) 

except TypeError: 

raise TypeError("The argument must be of type Composition") 

 

cpl = [x for x in reversed(dc.conjugate())] 

res = [] 

s = 0 

for i in range(len(cpl)): 

res += [s + cpl[i]-j for j in range(cpl[i])] 

s += cpl[i] 

 

return Permutations()(res) 

 

def descents_composition_last(dc): 

r""" 

Return the largest element of a descent class having a descent 

composition ``dc``. 

 

EXAMPLES:: 

 

sage: import sage.combinat.permutation as permutation 

sage: permutation.descents_composition_last([1,1,3,4,3]) 

[12, 11, 8, 9, 10, 4, 5, 6, 7, 1, 2, 3] 

""" 

if not isinstance(dc, Composition): 

try: 

dc = Composition(dc) 

except TypeError: 

raise TypeError("The argument must be of type Composition") 

s = 0 

res = [] 

for i in reversed(range(len(dc))): 

res = [j for j in range(s+1,s+dc[i]+1)] + res 

s += dc[i] 

 

return Permutations()(res) 

 

class StandardPermutations_recoilsfiner(Permutations): 

@staticmethod 

def __classcall_private__(cls, recoils): 

""" 

Normalize arguments to ensure a unique representation. 

 

TESTS:: 

 

sage: S1 = Permutations(recoils_finer=[2,2]) 

sage: S2 = Permutations(recoils_finer=(2,2)) 

sage: S1 is S2 

True 

""" 

return super(StandardPermutations_recoilsfiner, cls).__classcall__(cls, Composition(recoils)) 

 

def __init__(self, recoils): 

""" 

TESTS:: 

 

sage: P = Permutations(recoils_finer=[2,2]) 

sage: TestSuite(P).run() 

""" 

Permutations.__init__(self, category=FiniteEnumeratedSets()) 

self.recoils = recoils 

 

def _repr_(self): 

""" 

TESTS:: 

 

sage: Permutations(recoils_finer=[2,2]) 

Standard permutations whose recoils composition is finer than [2, 2] 

""" 

return "Standard permutations whose recoils composition is finer than %s"%self.recoils 

 

def __iter__(self): 

""" 

Iterate over of all of the permutations whose recoils composition 

is finer than ``self.recoils``. 

 

EXAMPLES:: 

 

sage: Permutations(recoils_finer=[2,2]).list() 

[[1, 2, 3, 4], 

[1, 3, 2, 4], 

[1, 3, 4, 2], 

[3, 1, 2, 4], 

[3, 1, 4, 2], 

[3, 4, 1, 2]] 

""" 

recoils = self.recoils 

dag = DiGraph() 

 

#Add the nodes 

for i in range(1, sum(recoils)+1): 

dag.add_vertex(i) 

 

#Add the edges to guarantee a finer recoil composition 

pos = 1 

for part in recoils: 

for i in range(part-1): 

dag.add_edge(pos, pos+1) 

pos += 1 

pos += 1 

 

for le in dag.topological_sort_generator(): 

yield self.element_class(self, le) 

 

class StandardPermutations_recoilsfatter(Permutations): 

@staticmethod 

def __classcall_private__(cls, recoils): 

""" 

Normalize arguments to ensure a unique representation. 

 

TESTS:: 

 

sage: S1 = Permutations(recoils_fatter=[2,2]) 

sage: S2 = Permutations(recoils_fatter=(2,2)) 

sage: S1 is S2 

True 

""" 

return super(StandardPermutations_recoilsfatter, cls).__classcall__(cls, Composition(recoils)) 

 

def __init__(self, recoils): 

""" 

TESTS:: 

 

sage: P = Permutations(recoils_fatter=[2,2]) 

sage: TestSuite(P).run() 

""" 

Permutations.__init__(self, category=FiniteEnumeratedSets()) 

self.recoils = recoils 

 

def _repr_(self): 

""" 

TESTS:: 

 

sage: Permutations(recoils_fatter=[2,2]) 

Standard permutations whose recoils composition is fatter than [2, 2] 

""" 

return "Standard permutations whose recoils composition is fatter than %s"%self.recoils 

 

def __iter__(self): 

""" 

Iterate over of all of the permutations whose recoils composition 

is fatter than ``self.recoils``. 

 

EXAMPLES:: 

 

sage: Permutations(recoils_fatter=[2,2]).list() 

[[1, 3, 2, 4], 

[1, 3, 4, 2], 

[1, 4, 3, 2], 

[3, 1, 2, 4], 

[3, 1, 4, 2], 

[3, 2, 1, 4], 

[3, 2, 4, 1], 

[3, 4, 1, 2], 

[3, 4, 2, 1], 

[4, 1, 3, 2], 

[4, 3, 1, 2], 

[4, 3, 2, 1]] 

""" 

recoils = self.recoils 

dag = DiGraph() 

 

#Add the nodes 

for i in range(1, sum(recoils)+1): 

dag.add_vertex(i) 

 

#Add the edges to guarantee a fatter recoil composition 

pos = 0 

for i in range(len(recoils)-1): 

pos += recoils[i] 

dag.add_edge(pos+1, pos) 

 

for le in dag.topological_sort_generator(): 

yield self.element_class(self, le) 

 

class StandardPermutations_recoils(Permutations): 

r""" 

Permutations of `\{1, \ldots, n\}` with a fixed recoils composition. 

""" 

@staticmethod 

def __classcall_private__(cls, recoils): 

""" 

Normalize arguments to ensure a unique representation. 

 

TESTS:: 

 

sage: S1 = Permutations(recoils=[2,2]) 

sage: S2 = Permutations(recoils=(2,2)) 

sage: S1 is S2 

True 

""" 

return super(StandardPermutations_recoils, cls).__classcall__(cls, Composition(recoils)) 

 

def __init__(self, recoils): 

""" 

TESTS:: 

 

sage: P = Permutations(recoils=[2,2]) 

sage: TestSuite(P).run() 

""" 

Permutations.__init__(self, category=FiniteEnumeratedSets()) 

self.recoils = recoils 

 

def _repr_(self): 

""" 

TESTS:: 

 

sage: Permutations(recoils=[2,2]) 

Standard permutations whose recoils composition is [2, 2] 

""" 

return "Standard permutations whose recoils composition is %s"%self.recoils 

 

def __iter__(self): 

""" 

Iterate over of all of the permutations whose recoils composition 

is equal to ``self.recoils``. 

 

EXAMPLES:: 

 

sage: Permutations(recoils=[2,2]).list() 

[[1, 3, 2, 4], [1, 3, 4, 2], [3, 1, 2, 4], [3, 1, 4, 2], [3, 4, 1, 2]] 

""" 

recoils = self.recoils 

dag = DiGraph() 

 

#Add all the nodes 

for i in range(1, sum(recoils)+1): 

dag.add_vertex(i) 

 

#Add the edges which guarantee a finer recoil comp. 

pos = 1 

for part in recoils: 

for i in range(part-1): 

dag.add_edge(pos, pos+1) 

pos += 1 

pos += 1 

 

#Add the edges which guarantee a fatter recoil comp. 

pos = 0 

for i in range(len(recoils)-1): 

pos += recoils[i] 

dag.add_edge(pos+1, pos) 

 

for le in dag.topological_sort_generator(): 

yield self.element_class(self, le) 

 

def from_major_code(mc, final_descent=False): 

r""" 

Return the permutation with major code ``mc``. 

 

The major code of a permutation is defined in 

:meth:`~sage.combinat.permutation.Permutation.to_major_code`. 

 

.. WARNING:: 

 

This function creates illegal permutations (i.e. ``Permutation([9])``, 

and this is dangerous as the :meth:`Permutation` class is only designed 

to handle permutations on `1...n`. This will have to be changed when Sage 

permutations will be able to handle anything, but right now this should 

be fixed. Be careful with the results. 

 

.. WARNING:: 

 

If ``mc`` is not a major index of a permutation, then the return 

value of this method can be anything. Garbage in, garbage out! 

 

REFERENCES: 

 

- Skandera, M. *An Eulerian Partner for Inversions*. Sem. 

Lothar. Combin. 46 (2001) B46d. 

 

EXAMPLES:: 

 

sage: import sage.combinat.permutation as permutation 

sage: permutation.from_major_code([5, 0, 1, 0, 1, 2, 0, 1, 0]) 

[9, 3, 5, 7, 2, 1, 4, 6, 8] 

sage: permutation.from_major_code([8, 3, 3, 1, 4, 0, 1, 0, 0]) 

[2, 8, 4, 3, 6, 7, 9, 5, 1] 

sage: Permutation([2,1,6,4,7,3,5]).to_major_code() 

[3, 2, 0, 2, 2, 0, 0] 

sage: permutation.from_major_code([3, 2, 0, 2, 2, 0, 0]) 

[2, 1, 6, 4, 7, 3, 5] 

 

TESTS:: 

 

sage: permutation.from_major_code([]) 

[] 

 

sage: all( permutation.from_major_code(p.to_major_code()) == p 

....: for p in Permutations(5) ) 

True 

""" 

if not mc: 

w = [] 

else: 

#define w^(n) to be the one-letter word n 

w = [len(mc)] 

 

#for i=n-1,..,1 let w^i be the unique word obtained by inserting 

#the letter i into the word w^(i+1) in such a way that 

#maj(w^i)-maj(w^(i+1)) = mc[i] 

 

for i in reversed(range(1,len(mc))): 

#Lemma 2.2 in Skandera 

 

#Get the descents of w and place them in reverse order 

d = Permutation(w, check_input=False).descents(final_descent=final_descent) 

d.reverse() 

 

#a is the list of all positions which are not descents 

a = [x for x in range(1, len(w) + 1) if x not in d] 

 

#d_k = -1 -- 0 in the lemma, but -1 due to 0-based indexing 

d.append(0) 

l = mc[i-1] 

indices = d + a 

w.insert(indices[l], i) 

 

return Permutation(w, check_input = False) 

 

################ 

# Bruhat Order # 

################ 

 

 

class StandardPermutations_bruhat_smaller(Permutations): 

r""" 

Permutations of `\{1, \ldots, n\}` that are less than or equal to a 

permutation `p` in the Bruhat order. 

""" 

@staticmethod 

def __classcall_private__(cls, p): 

""" 

Normalize arguments to ensure a unique representation. 

 

TESTS:: 

 

sage: S1 = Permutations(bruhat_smaller=[2,3,1]) 

sage: S2 = Permutations(bruhat_smaller=(1,2,3)) 

sage: S1 is S2 

True 

""" 

return super(StandardPermutations_bruhat_smaller, cls).__classcall__(cls, Permutation(p)) 

 

def __init__(self, p): 

""" 

TESTS:: 

 

sage: P = Permutations(bruhat_smaller=[3,2,1]) 

sage: TestSuite(P).run() 

""" 

Permutations.__init__(self, category=FiniteEnumeratedSets()) 

self.p = p 

 

def _repr_(self): 

""" 

TESTS:: 

 

sage: Permutations(bruhat_smaller=[3,2,1]) 

Standard permutations that are less than or equal to [3, 2, 1] in the Bruhat order 

""" 

return "Standard permutations that are less than or equal to %s in the Bruhat order"%self.p 

 

def __iter__(self): 

r""" 

Iterate through a list of permutations smaller than or equal to ``p`` 

in the Bruhat order. 

 

EXAMPLES:: 

 

sage: Permutations(bruhat_smaller=[4,1,2,3]).list() 

[[1, 2, 3, 4], 

[1, 2, 4, 3], 

[1, 3, 2, 4], 

[1, 4, 2, 3], 

[2, 1, 3, 4], 

[2, 1, 4, 3], 

[3, 1, 2, 4], 

[4, 1, 2, 3]] 

""" 

return iter(transitive_ideal(lambda x: x.bruhat_pred(), self.p)) 

 

 

class StandardPermutations_bruhat_greater(Permutations): 

r""" 

Permutations of `\{1, \ldots, n\}` that are greater than or equal to a 

permutation `p` in the Bruhat order. 

""" 

@staticmethod 

def __classcall_private__(cls, p): 

""" 

Normalize arguments to ensure a unique representation. 

 

TESTS:: 

 

sage: S1 = Permutations(bruhat_greater=[2,3,1]) 

sage: S2 = Permutations(bruhat_greater=(1,2,3)) 

sage: S1 is S2 

True 

""" 

return super(StandardPermutations_bruhat_greater, cls).__classcall__(cls, Permutation(p)) 

 

def __init__(self, p): 

""" 

TESTS:: 

 

sage: P = Permutations(bruhat_greater=[3,2,1]) 

sage: TestSuite(P).run() 

""" 

Permutations.__init__(self, category=FiniteEnumeratedSets()) 

self.p = p 

 

def _repr_(self): 

""" 

TESTS:: 

 

sage: Permutations(bruhat_greater=[3,2,1]) 

Standard permutations that are greater than or equal to [3, 2, 1] in the Bruhat order 

""" 

return "Standard permutations that are greater than or equal to %s in the Bruhat order"%self.p 

 

def __iter__(self): 

r""" 

Iterate through a list of permutations greater than or equal to ``p`` 

in the Bruhat order. 

 

EXAMPLES:: 

 

sage: Permutations(bruhat_greater=[4,1,2,3]).list() 

[[4, 1, 2, 3], 

[4, 1, 3, 2], 

[4, 2, 1, 3], 

[4, 2, 3, 1], 

[4, 3, 1, 2], 

[4, 3, 2, 1]] 

""" 

return iter(transitive_ideal(lambda x: x.bruhat_succ(), self.p)) 

 

def bruhat_lequal(p1, p2): 

r""" 

Return ``True`` if ``p1`` is less than ``p2`` in the Bruhat order. 

 

Algorithm from mupad-combinat. 

 

EXAMPLES:: 

 

sage: import sage.combinat.permutation as permutation 

sage: permutation.bruhat_lequal([2,4,3,1],[3,4,2,1]) 

True 

""" 

n1 = len(p1) 

 

if n1 == 0: 

return True 

 

if p1[0] > p2[0] or p1[n1-1] < p2[n1-1]: 

return False 

 

for i in range(n1): 

c = 0 

for j in range(n1): 

if p2[j] > i+1: 

c += 1 

if p1[j] > i+1: 

c -= 1 

if c < 0: 

return False 

 

return True 

 

 

################# 

# Permutohedron # 

################# 

 

def permutohedron_lequal(p1, p2, side="right"): 

r""" 

Return ``True`` if ``p1`` is less than or equal to ``p2`` in the 

permutohedron order. 

 

By default, the computations are done in the right permutohedron. 

If you pass the option ``side='left'``, then they will be done in the 

left permutohedron. 

 

EXAMPLES:: 

 

sage: import sage.combinat.permutation as permutation 

sage: permutation.permutohedron_lequal(Permutation([3,2,1,4]),Permutation([4,2,1,3])) 

False 

sage: permutation.permutohedron_lequal(Permutation([3,2,1,4]),Permutation([4,2,1,3]), side='left') 

True 

""" 

l1 = p1.number_of_inversions() 

l2 = p2.number_of_inversions() 

 

if l1 > l2: 

return False 

 

if side == "right": 

prod = p1._left_to_right_multiply_on_right(p2.inverse()) 

else: 

prod = p1._left_to_right_multiply_on_left(p2.inverse()) 

 

return prod.number_of_inversions() == l2 - l1 

 

 

############ 

# Patterns # 

############ 

from sage.combinat.words.finite_word import evaluation_dict 

 

def to_standard(p, key=None): 

r""" 

Return a standard permutation corresponding to the iterable ``p``. 

 

INPUT: 

 

- ``p`` -- an iterable 

- ``key`` -- (optional) a comparison key for the element 

``x`` of ``p`` 

 

EXAMPLES:: 

 

sage: import sage.combinat.permutation as permutation 

sage: permutation.to_standard([4,2,7]) 

[2, 1, 3] 

sage: permutation.to_standard([1,2,3]) 

[1, 2, 3] 

sage: permutation.to_standard([]) 

[] 

sage: permutation.to_standard([1,2,3], key=lambda x: -x) 

[3, 2, 1] 

sage: permutation.to_standard([5,8,2,5], key=lambda x: -x) 

[2, 1, 4, 3] 

 

TESTS: 

 

Does not mutate the list:: 

 

sage: a = [1,2,4] 

sage: permutation.to_standard(a) 

[1, 2, 3] 

sage: a 

[1, 2, 4] 

 

We check against the naive method:: 

 

sage: def std(p): 

....: s = [0]*len(p) 

....: c = p[:] 

....: biggest = max(p) + 1 

....: i = 1 

....: for _ in range(len(c)): 

....: smallest = min(c) 

....: smallest_index = c.index(smallest) 

....: s[smallest_index] = i 

....: i += 1 

....: c[smallest_index] = biggest 

....: return Permutations()(s) 

sage: p = list(Words(100, 1000).random_element()) 

sage: std(p) == permutation.to_standard(p) 

True 

""" 

ev_dict = evaluation_dict(p) 

ordered_alphabet = sorted(ev_dict, key=key) 

offset = 0 

for k in ordered_alphabet: 

temp = ev_dict[k] 

ev_dict[k] = offset 

offset += temp 

result = [] 

for l in p: 

ev_dict[l] += 1 

result.append(ev_dict[l]) 

return Permutations(len(result))(result) 

 

 

########################################################## 

 

class CyclicPermutations(Permutations_mset): 

""" 

Return the class of all cyclic permutations of ``mset`` in cycle notation. 

These are the same as necklaces. 

 

INPUT: 

 

- ``mset`` -- A multiset 

 

EXAMPLES:: 

 

sage: CyclicPermutations(range(4)).list() 

[[0, 1, 2, 3], 

[0, 1, 3, 2], 

[0, 2, 1, 3], 

[0, 2, 3, 1], 

[0, 3, 1, 2], 

[0, 3, 2, 1]] 

sage: CyclicPermutations([1,1,1]).list() 

[[1, 1, 1]] 

""" 

@staticmethod 

def __classcall_private__(cls, mset): 

""" 

Normalize arguments to ensure a unique representation. 

 

TESTS:: 

 

sage: CP1 = CyclicPermutations([1,1,1]) 

sage: CP2 = CyclicPermutations((1,1,1)) 

sage: CP1 is CP2 

True 

 

sage: CP = CyclicPermutations([1,2,3,3]) 

sage: CP 

Cyclic permutations of [1, 2, 3, 3] 

sage: TestSuite(CP).run() # not tested -- broken 

""" 

return super(CyclicPermutations, cls).__classcall__(cls, tuple(mset)) 

 

def _repr_(self): 

""" 

TESTS:: 

 

sage: CyclicPermutations(range(4)) 

Cyclic permutations of [0, 1, 2, 3] 

""" 

return "Cyclic permutations of %s"%list(self.mset) 

 

def __iter__(self, distinct=False): 

""" 

EXAMPLES:: 

 

sage: CyclicPermutations(range(4)).list() # indirect doctest 

[[0, 1, 2, 3], 

[0, 1, 3, 2], 

[0, 2, 1, 3], 

[0, 2, 3, 1], 

[0, 3, 1, 2], 

[0, 3, 2, 1]] 

sage: CyclicPermutations([1,1,1]).list() 

[[1, 1, 1]] 

sage: CyclicPermutations([1,1,1]).list(distinct=True) 

[[1, 1, 1], [1, 1, 1]] 

""" 

if distinct: 

content = [1] * len(self.mset) 

else: 

content = [0] * len(self.mset) 

index_list = map(self.mset.index, self.mset) 

for i in index_list: 

content[i] += 1 

 

from .necklace import Necklaces 

for necklace in Necklaces(content): 

yield [self.mset[x-1] for x in necklace] 

 

iterator = __iter__ 

 

def list(self, distinct=False): 

""" 

EXAMPLES:: 

 

sage: CyclicPermutations(range(4)).list() 

[[0, 1, 2, 3], 

[0, 1, 3, 2], 

[0, 2, 1, 3], 

[0, 2, 3, 1], 

[0, 3, 1, 2], 

[0, 3, 2, 1]] 

""" 

return list(self.__iter__(distinct=distinct)) 

 

################################################# 

 

class CyclicPermutationsOfPartition(Permutations): 

""" 

Combinations of cyclic permutations of each cell of a given partition. 

 

This is the same as a Cartesian product of necklaces. 

 

EXAMPLES:: 

 

sage: CyclicPermutationsOfPartition([[1,2,3,4],[5,6,7]]).list() 

[[[1, 2, 3, 4], [5, 6, 7]], 

[[1, 2, 4, 3], [5, 6, 7]], 

[[1, 3, 2, 4], [5, 6, 7]], 

[[1, 3, 4, 2], [5, 6, 7]], 

[[1, 4, 2, 3], [5, 6, 7]], 

[[1, 4, 3, 2], [5, 6, 7]], 

[[1, 2, 3, 4], [5, 7, 6]], 

[[1, 2, 4, 3], [5, 7, 6]], 

[[1, 3, 2, 4], [5, 7, 6]], 

[[1, 3, 4, 2], [5, 7, 6]], 

[[1, 4, 2, 3], [5, 7, 6]], 

[[1, 4, 3, 2], [5, 7, 6]]] 

 

:: 

 

sage: CyclicPermutationsOfPartition([[1,2,3,4],[4,4,4]]).list() 

[[[1, 2, 3, 4], [4, 4, 4]], 

[[1, 2, 4, 3], [4, 4, 4]], 

[[1, 3, 2, 4], [4, 4, 4]], 

[[1, 3, 4, 2], [4, 4, 4]], 

[[1, 4, 2, 3], [4, 4, 4]], 

[[1, 4, 3, 2], [4, 4, 4]]] 

 

:: 

 

sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list() 

[[[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]]] 

 

:: 

 

sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list(distinct=True) 

[[[1, 2, 3], [4, 4, 4]], 

[[1, 3, 2], [4, 4, 4]], 

[[1, 2, 3], [4, 4, 4]], 

[[1, 3, 2], [4, 4, 4]]] 

""" 

@staticmethod 

def __classcall_private__(cls, partition): 

""" 

Normalize arguments to ensure a unique representation. 

 

TESTS:: 

 

sage: CP1 = CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]) 

sage: CP2 = CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]) 

sage: CP1 is CP2 

True 

""" 

partition = tuple(map(tuple, partition)) 

return super(CyclicPermutationsOfPartition, cls).__classcall__(cls, partition) 

 

def __init__(self, partition): 

""" 

TESTS:: 

 

sage: CP = CyclicPermutationsOfPartition([[1,2,3,4],[5,6,7]]) 

sage: CP 

Cyclic permutations of partition [[1, 2, 3, 4], [5, 6, 7]] 

sage: TestSuite(CP).run() 

""" 

self.partition = partition 

Permutations.__init__(self, category=FiniteEnumeratedSets()) 

 

class Element(ClonableArray): 

""" 

A cyclic permutation of a partition. 

""" 

def check(self): 

""" 

Check that ``self`` is a valid element. 

 

EXAMPLES:: 

 

sage: CP = CyclicPermutationsOfPartition([[1,2,3,4],[5,6,7]]) 

sage: elt = CP[0] 

sage: elt.check() 

""" 

if [sorted(_) for _ in self] != [sorted(_) for _ in self.parent().partition]: 

raise ValueError("Invalid cyclic permutation of the partition"%self.parent().partition) 

 

def _repr_(self): 

""" 

TESTS:: 

 

sage: CyclicPermutationsOfPartition([[1,2,3,4],[5,6,7]]) 

Cyclic permutations of partition [[1, 2, 3, 4], [5, 6, 7]] 

""" 

return "Cyclic permutations of partition {}".format( 

[list(_) for _ in self.partition]) 

 

def __iter__(self, distinct=False): 

""" 

AUTHORS: 

 

- Robert Miller 

 

EXAMPLES:: 

 

sage: CyclicPermutationsOfPartition([[1,2,3,4],[5,6,7]]).list() # indirect doctest 

[[[1, 2, 3, 4], [5, 6, 7]], 

[[1, 2, 4, 3], [5, 6, 7]], 

[[1, 3, 2, 4], [5, 6, 7]], 

[[1, 3, 4, 2], [5, 6, 7]], 

[[1, 4, 2, 3], [5, 6, 7]], 

[[1, 4, 3, 2], [5, 6, 7]], 

[[1, 2, 3, 4], [5, 7, 6]], 

[[1, 2, 4, 3], [5, 7, 6]], 

[[1, 3, 2, 4], [5, 7, 6]], 

[[1, 3, 4, 2], [5, 7, 6]], 

[[1, 4, 2, 3], [5, 7, 6]], 

[[1, 4, 3, 2], [5, 7, 6]]] 

 

:: 

 

sage: CyclicPermutationsOfPartition([[1,2,3,4],[4,4,4]]).list() 

[[[1, 2, 3, 4], [4, 4, 4]], 

[[1, 2, 4, 3], [4, 4, 4]], 

[[1, 3, 2, 4], [4, 4, 4]], 

[[1, 3, 4, 2], [4, 4, 4]], 

[[1, 4, 2, 3], [4, 4, 4]], 

[[1, 4, 3, 2], [4, 4, 4]]] 

 

:: 

 

sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list() 

[[[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]]] 

 

:: 

 

sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list(distinct=True) 

[[[1, 2, 3], [4, 4, 4]], 

[[1, 3, 2], [4, 4, 4]], 

[[1, 2, 3], [4, 4, 4]], 

[[1, 3, 2], [4, 4, 4]]] 

""" 

if len(self.partition) == 1: 

for i in CyclicPermutations(self.partition[0]).iterator(distinct=distinct): 

yield self.element_class(self, [i]) 

else: 

for right in CyclicPermutationsOfPartition(self.partition[1:]).iterator(distinct=distinct): 

for perm in CyclicPermutations(self.partition[0]).iterator(distinct=distinct): 

yield self.element_class(self, [perm] + list(right)) 

 

iterator = __iter__ 

 

def list(self, distinct=False): 

""" 

EXAMPLES:: 

 

sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list() 

[[[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]]] 

sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list(distinct=True) 

[[[1, 2, 3], [4, 4, 4]], 

[[1, 3, 2], [4, 4, 4]], 

[[1, 2, 3], [4, 4, 4]], 

[[1, 3, 2], [4, 4, 4]]] 

""" 

return list(self.iterator(distinct=distinct)) 

 

 

############################################### 

#Avoiding 

 

class StandardPermutations_avoiding_generic(StandardPermutations_n_abstract): 

""" 

Generic class for subset of permutations avoiding a particular pattern. 

""" 

@staticmethod 

def __classcall_private__(cls, n, a): 

""" 

Normalize arguments to ensure a unique representation. 

 

TESTS:: 

 

sage: P1 = Permutations(3, avoiding=([2, 1, 3],[1,2,3])) 

sage: P2 = Permutations(3, avoiding=[[2, 1, 3],[1,2,3]]) 

sage: P1 is P2 

True 

""" 

a = tuple(map(Permutation, a)) 

return super(StandardPermutations_avoiding_generic, cls).__classcall__(cls, n, a) 

 

def __init__(self, n, a): 

""" 

EXAMPLES:: 

 

sage: P = Permutations(3, avoiding=[[2, 1, 3],[1,2,3]]) 

sage: TestSuite(P).run() 

sage: type(P) 

<class 'sage.combinat.permutation.StandardPermutations_avoiding_generic_with_category'> 

""" 

StandardPermutations_n_abstract.__init__(self, n) 

self.a = a 

 

def _repr_(self): 

""" 

EXAMPLES:: 

 

sage: Permutations(3, avoiding=[[2, 1, 3],[1,2,3]]) 

Standard permutations of 3 avoiding [[2, 1, 3], [1, 2, 3]] 

""" 

return "Standard permutations of %s avoiding %s"%(self.n, list(self.a)) 

 

def __iter__(self): 

""" 

EXAMPLES:: 

 

sage: Permutations(3, avoiding=[[2, 1, 3],[1,2,3]]).list() 

[[1, 3, 2], [3, 1, 2], [2, 3, 1], [3, 2, 1]] 

sage: Permutations(0, avoiding=[[2, 1, 3],[1,2,3]]).list() 

[[]] 

""" 

if self.n > 0: 

return iter(PatternAvoider(self, self.a)) 

return iter([self.element_class(self, [])]) 

 

def cardinality(self): 

""" 

Return the cardinality of ``self``. 

 

EXAMPLES:: 

 

sage: P = Permutations(3, avoiding=[[2, 1, 3],[1,2,3]]) 

sage: P.cardinality() 

4 

""" 

one = ZZ.one() 

return sum(one for p in self) 

 

class StandardPermutations_avoiding_12(StandardPermutations_avoiding_generic): 

def __init__(self, n): 

""" 

TESTS:: 

 

sage: P = Permutations(3, avoiding=[1, 2]) 

sage: TestSuite(P).run() 

""" 

StandardPermutations_avoiding_generic.__init__(self, n, Permutations()([1, 2])) 

 

def __iter__(self): 

""" 

EXAMPLES:: 

 

sage: Permutations(3, avoiding=[1,2]).list() 

[[3, 2, 1]] 

""" 

yield self.element_class(self, range(self.n, 0, -1)) 

 

def cardinality(self): 

""" 

Return the cardinality of ``self``. 

 

EXAMPLES:: 

 

sage: P = Permutations(3, avoiding=[1, 2]) 

sage: P.cardinality() 

1 

""" 

return ZZ.one() 

 

class StandardPermutations_avoiding_21(StandardPermutations_avoiding_generic): 

def __init__(self, n): 

""" 

TESTS:: 

 

sage: P = Permutations(3, avoiding=[2, 1]) 

sage: TestSuite(P).run() 

""" 

StandardPermutations_avoiding_generic.__init__(self, n, Permutations()([2, 1])) 

 

def __iter__(self): 

""" 

EXAMPLES:: 

 

sage: Permutations(3, avoiding=[2,1]).list() 

[[1, 2, 3]] 

""" 

yield self.element_class(self, range(1, self.n+1)) 

 

def cardinality(self): 

""" 

Return the cardinality of ``self``. 

 

EXAMPLES:: 

 

sage: P = Permutations(3, avoiding=[2, 1]) 

sage: P.cardinality() 

1 

""" 

return ZZ.one() 

 

class StandardPermutations_avoiding_132(StandardPermutations_avoiding_generic): 

def __init__(self, n): 

""" 

TESTS:: 

 

sage: P = Permutations(3, avoiding=[1, 3, 2]) 

sage: TestSuite(P).run() 

""" 

StandardPermutations_avoiding_generic.__init__(self, n, Permutations()([1, 3, 2])) 

 

def cardinality(self): 

""" 

EXAMPLES:: 

 

sage: Permutations(5, avoiding=[1, 3, 2]).cardinality() 

42 

sage: len( Permutations(5, avoiding=[1, 3, 2]).list() ) 

42 

""" 

return catalan_number(self.n) 

 

def __iter__(self): 

""" 

EXAMPLES:: 

 

sage: Permutations(3, avoiding=[1,3,2]).list() # indirect doctest 

[[1, 2, 3], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]] 

sage: Permutations(4, avoiding=[1,3,2]).list() 

[[4, 1, 2, 3], 

[4, 2, 1, 3], 

[4, 2, 3, 1], 

[4, 3, 1, 2], 

[4, 3, 2, 1], 

[3, 4, 1, 2], 

[3, 4, 2, 1], 

[2, 3, 4, 1], 

[3, 2, 4, 1], 

[1, 2, 3, 4], 

[2, 1, 3, 4], 

[2, 3, 1, 4], 

[3, 1, 2, 4], 

[3, 2, 1, 4]] 

""" 

if self.n == 0: 

return 

 

elif self.n < 3: 

for p in itertools.permutations(range(1, self.n + 1)): 

yield self.element_class(self, p) 

return 

 

elif self.n == 3: 

for p in itertools.permutations(range(1, self.n + 1)): 

if p != (1, 3, 2): 

yield self.element_class(self, p) 

return 

 

#Yield all the 132 avoiding permutations to the right. 

for right in StandardPermutations_avoiding_132(self.n - 1): 

yield self.element_class(self, [self.n] + list(right)) 

 

#yi 

for i in range(1, self.n-1): 

for left in StandardPermutations_avoiding_132(i): 

for right in StandardPermutations_avoiding_132(self.n-i-1): 

yield self.element_class(self, [x+(self.n-i-1) for x in left] + [self.n] + list(right) ) 

 

 

#Yield all the 132 avoiding permutations to the left 

for left in StandardPermutations_avoiding_132(self.n - 1): 

yield self.element_class(self, list(left) + [self.n]) 

 

class StandardPermutations_avoiding_123(StandardPermutations_avoiding_generic): 

def __init__(self, n): 

""" 

TESTS:: 

 

sage: P = Permutations(3, avoiding=[2, 1, 3]) 

sage: TestSuite(P).run() 

""" 

StandardPermutations_avoiding_generic.__init__(self, n, Permutations()([1, 2, 3])) 

 

def cardinality(self): 

""" 

EXAMPLES:: 

 

sage: Permutations(5, avoiding=[1, 2, 3]).cardinality() 

42 

sage: len( Permutations(5, avoiding=[1, 2, 3]).list() ) 

42 

""" 

return catalan_number(self.n) 

 

def __iter__(self): 

""" 

EXAMPLES:: 

 

sage: Permutations(3, avoiding=[1, 2, 3]).list() # indirect doctest 

[[1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]] 

sage: Permutations(2, avoiding=[1, 2, 3]).list() 

[[1, 2], [2, 1]] 

sage: Permutations(3, avoiding=[1, 2, 3]).list() 

[[1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]] 

""" 

if self.n == 0: 

return 

 

elif self.n < 3: 

for p in itertools.permutations(range(1, self.n + 1)): 

yield self.element_class(self, p) 

return 

 

elif self.n == 3: 

for p in itertools.permutations(range(1, self.n + 1)): 

if p != (1, 2, 3): 

yield self.element_class(self, p) 

return 

 

for p in StandardPermutations_avoiding_132(self.n): 

#Convert p to a 123 avoiding permutation by 

m = self.n+1 

minima_pos = [] 

minima = [] 

for i in range(self.n): 

if p[i] < m: 

minima_pos.append(i) 

minima.append(p[i]) 

m = p[i] 

 

new_p = [] 

non_minima = [x for x in range(self.n, 0, -1) if x not in minima] 

a = 0 

b = 0 

for i in range(self.n): 

if i in minima_pos: 

new_p.append( minima[a] ) 

a += 1 

else: 

new_p.append( non_minima[b] ) 

b += 1 

 

yield self.element_class(self, new_p) 

 

class StandardPermutations_avoiding_321(StandardPermutations_avoiding_generic): 

def __init__(self, n): 

""" 

TESTS:: 

 

sage: P = Permutations(3, avoiding=[3, 2, 1]) 

sage: TestSuite(P).run() 

""" 

StandardPermutations_avoiding_generic.__init__(self, n, Permutations()([3, 2, 1])) 

 

def cardinality(self): 

""" 

EXAMPLES:: 

 

sage: Permutations(5, avoiding=[3, 2, 1]).cardinality() 

42 

sage: len( Permutations(5, avoiding=[3, 2, 1]).list() ) 

42 

""" 

return catalan_number(self.n) 

 

def __iter__(self): 

""" 

EXAMPLES:: 

 

sage: Permutations(3, avoiding=[3, 2, 1]).list() #indirect doctest 

[[2, 3, 1], [3, 1, 2], [1, 3, 2], [2, 1, 3], [1, 2, 3]] 

""" 

for p in StandardPermutations_avoiding_123(self.n): 

yield self.element_class(self, p.reverse()) 

 

class StandardPermutations_avoiding_231(StandardPermutations_avoiding_generic): 

def __init__(self, n): 

""" 

TESTS:: 

 

sage: P = Permutations(3, avoiding=[2, 3, 1]) 

sage: TestSuite(P).run() 

""" 

StandardPermutations_avoiding_generic.__init__(self, n, Permutations()([2, 3, 1])) 

 

def cardinality(self): 

""" 

EXAMPLES:: 

 

sage: Permutations(5, avoiding=[2, 3, 1]).cardinality() 

42 

sage: len( Permutations(5, avoiding=[2, 3, 1]).list() ) 

42 

""" 

return catalan_number(self.n) 

 

def __iter__(self): 

""" 

EXAMPLES:: 

 

sage: Permutations(3, avoiding=[2, 3, 1]).list() 

[[3, 2, 1], [3, 1, 2], [1, 3, 2], [2, 1, 3], [1, 2, 3]] 

""" 

for p in StandardPermutations_avoiding_132(self.n): 

yield self.element_class(self, p.reverse()) 

 

 

class StandardPermutations_avoiding_312(StandardPermutations_avoiding_generic): 

def __init__(self, n): 

""" 

TESTS:: 

 

sage: P = Permutations(3, avoiding=[3, 1, 2]) 

sage: TestSuite(P).run() 

""" 

super(StandardPermutations_avoiding_312, self).__init__(n, Permutations()([3, 1, 2])) 

 

def cardinality(self): 

""" 

EXAMPLES:: 

 

sage: Permutations(5, avoiding=[3, 1, 2]).cardinality() 

42 

sage: len( Permutations(5, avoiding=[3, 1, 2]).list() ) 

42 

""" 

return catalan_number(self.n) 

 

def __iter__(self): 

""" 

EXAMPLES:: 

 

sage: Permutations(3, avoiding=[3, 1, 2]).list() 

[[3, 2, 1], [2, 3, 1], [2, 1, 3], [1, 3, 2], [1, 2, 3]] 

""" 

for p in StandardPermutations_avoiding_132(self.n): 

yield self.element_class(self, p.complement()) 

 

 

class StandardPermutations_avoiding_213(StandardPermutations_avoiding_generic): 

def __init__(self, n): 

""" 

TESTS:: 

 

sage: P = Permutations(3, avoiding=[2, 1, 3]) 

sage: TestSuite(P).run() 

""" 

super(StandardPermutations_avoiding_213, self).__init__(n, Permutations()([2, 1, 3])) 

 

def cardinality(self): 

""" 

EXAMPLES:: 

 

sage: Permutations(5, avoiding=[2, 1, 3]).cardinality() 

42 

sage: len( Permutations(5, avoiding=[2, 1, 3]).list() ) 

42 

""" 

return catalan_number(self.n) 

 

def __iter__(self): 

""" 

EXAMPLES:: 

 

sage: Permutations(3, avoiding=[2, 1, 3]).list() 

[[1, 2, 3], [1, 3, 2], [3, 1, 2], [2, 3, 1], [3, 2, 1]] 

""" 

for p in StandardPermutations_avoiding_132(self.n): 

yield p.complement().reverse() 

 

 

class PatternAvoider(GenericBacktracker): 

def __init__(self, parent, patterns): 

""" 

EXAMPLES:: 

 

sage: from sage.combinat.permutation import PatternAvoider 

sage: P = Permutations(4) 

sage: p = PatternAvoider(P, [[1,2,3]]) 

sage: loads(dumps(p)) 

<sage.combinat.permutation.PatternAvoider object at 0x...> 

""" 

GenericBacktracker.__init__(self, [], 1) 

self._patterns = patterns 

self._parent = parent 

 

def _rec(self, obj, state): 

""" 

EXAMPLES:: 

 

sage: from sage.combinat.permutation import PatternAvoider 

sage: P = Permutations(4) 

sage: p = PatternAvoider(P, [[1,2]]) 

sage: list(p._rec([1], 2)) 

[([2, 1], 3, False)] 

""" 

i = state 

 

if state != self._parent.n: 

new_state = state + 1 

yld = False 

else: 

new_state = None 

yld = True 

 

for pos in reversed(range(len(obj)+1)): 

new_obj = self._parent.element_class(self._parent, obj[:pos] + [i] + obj[pos:]) 

if all( not new_obj.has_pattern(p) for p in self._patterns): 

yield new_obj, new_state, yld 

 

 

class PermutationsNK(Permutations_setk): 

""" 

This exists solely for unpickling ``PermutationsNK`` objects created 

with Sage <= 6.3. 

""" 

def __setstate__(self, state): 

r""" 

For unpickling old ``PermutationsNK`` objects. 

 

EXAMPLES:: 

 

sage: loads(b"x\x9cM\x90\xcdN\xc30\x10\x84\xd5B\x0bM\x81\xf2\xd3\x1ex" 

....: b"\x03\xb8\xe4\x80x\x8bJ\x16B\xf2y\xb5qV\xa9\x95\xd8\xce" 

....: b"\xda[$\x0eHp\xe0\xc0[\xe3\xb4j\xe1bi\xfd\xcd\x8cg\xfd96" 

....: b"\t\x1b*Mp\x95\xf5(eO\xd1m\x05\xc5\x06\x0f\xbe-^\xfe\xc6" 

....: b"\xa4\xd6\x05\x8f\x1e\xbfx\xfc\xc1'\x0f\xba\x00r\x15\xd5" 

....: b"\xb5\xf5\r\x9f*\xbd\x04\x13\xfc\x1bE\x01G\xb2\t5xt\xc4" 

....: b"\x13\xa5\xa7`j\x14\xe4\xa9\xd230(\xd4\x84\xf8\xceg\x03" 

....: b"\x18$\x89\xcf\x95\x1e\x83\xe7\xd9\xbeH\xccy\xa9\xb4>\xeb" 

....: b"(\x16\x0e[\x82\xc3\xc0\x85\x1e=\x7f\xbf\xf2\\\xcf\xa1!O" 

....: b"\x11%\xc4\xc4\x17\x83\xbf\xe5\xcbM\xc6O\x19_\xe9\tT\x98" 

....: b"\x88\x17J/\xa0\xb7\xa6\xed\x08r\xb3\x94w\xe0\xeb\xf5(W" 

....: b"\xa5\x8e\x1cy\x19*'\x89[\x93s\xf8F\xe9U~\xca\x8a\xc5\xee" 

....: b"\xb8Kg\x93\xf0\xad\xd2\xf7G\xcb\xa0\x80\x1eS\xcaG\xcc\x17" 

....: b"|\xf7\x93\x03\x0f>4\xbb\x8f\xdb\xd9\x96\xea\x1f0\x81\xa2" 

....: b"\xa1=X\xa9mU\xfe\x02=\xaa\x87\x14") 

Permutations of the set [0, 1, 2, 3] of length 2 

""" 

self.__class__ = Permutations_setk 

self.__init__(tuple(range(state['_n'])), state['_k']) 

 

from sage.structure.sage_object import register_unpickle_override 

register_unpickle_override("sage.combinat.permutation", "Permutation_class", Permutation) 

register_unpickle_override("sage.combinat.permutation", "CyclicPermutationsOfPartition_partition", CyclicPermutationsOfPartition) 

register_unpickle_override("sage.combinat.permutation", "CyclicPermutations_mset", CyclicPermutations) 

register_unpickle_override('sage.combinat.permutation_nk', 'PermutationsNK', PermutationsNK) 

 

# Deprecations from trac:18555. July 2016 

from sage.misc.superseded import deprecated_function_alias 

Permutations.global_options=deprecated_function_alias(18555, Permutations.options) 

PermutationOptions = deprecated_function_alias(18555, Permutations.options)