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""" 

A catalog of posets and lattices. 

 

Some common posets can be accessed through the ``posets.<tab>`` object:: 

 

sage: posets.PentagonPoset() 

Finite lattice containing 5 elements 

 

Moreover, the set of all posets of order `n` is represented by ``Posets(n)``:: 

 

sage: Posets(5) 

Posets containing 5 elements 

 

The infinite set of all posets can be used to find minimal examples:: 

 

sage: for P in Posets(): 

....: if not P.is_series_parallel(): 

....: break 

sage: P 

Finite poset containing 4 elements 

 

**Catalog of common posets:** 

 

.. csv-table:: 

:class: contentstable 

:widths: 30, 70 

:delim: | 

 

:meth:`~posets.AntichainPoset` | Return an antichain on `n` elements. 

:meth:`~posets.BooleanLattice` | Return the Boolean lattice on `2^n` elements. 

:meth:`~posets.ChainPoset` | Return a chain on `n` elements. 

:meth:`~posets.Crown` | Return the crown poset on `2n` elements. 

:meth:`~posets.DiamondPoset` | Return the lattice of rank two on `n` elements. 

:meth:`~posets.DivisorLattice` | Return the divisor lattice of an integer. 

:meth:`~posets.IntegerCompositions` | Return the poset of integer compositions of `n`. 

:meth:`~posets.IntegerPartitions` | Return the poset of integer partitions of ``n``. 

:meth:`~posets.IntegerPartitionsDominanceOrder` | Return the lattice of integer partitions on the integer `n` ordered by dominance. 

:meth:`~posets.NoncrossingPartitions` | Return the poset of noncrossing partitions of a finite Coxeter group ``W``. 

:meth:`~posets.PentagonPoset` | Return the Pentagon poset. 

:meth:`~posets.PowerPoset` | Return a power poset. 

:meth:`~posets.RandomLattice` | Return a random lattice on `n` elements. 

:meth:`~posets.RandomPoset` | Return a random poset on `n` elements. 

:meth:`~posets.RestrictedIntegerPartitions` | Return the poset of integer partitions of `n`, ordered by restricted refinement. 

:meth:`~posets.SetPartitions` | Return the poset of set partitions of the set `\{1,\dots,n\}`. 

:meth:`~posets.ShardPoset` | Return the shard intersection order. 

:meth:`~posets.SSTPoset` | Return the poset on semistandard tableaux of shape `s` and largest entry `f` that is ordered by componentwise comparison. 

:meth:`~posets.StandardExample` | Return the standard example of a poset with dimension `n`. 

:meth:`~posets.SymmetricGroupAbsoluteOrderPoset` | The poset of permutations with respect to absolute order. 

:meth:`~posets.SymmetricGroupBruhatIntervalPoset` | The poset of permutations with respect to Bruhat order. 

:meth:`~posets.SymmetricGroupBruhatOrderPoset` | The poset of permutations with respect to Bruhat order. 

:meth:`~posets.SymmetricGroupWeakOrderPoset` | The poset of permutations of `\{ 1, 2, \ldots, n \}` with respect to the weak order. 

:meth:`~posets.TamariLattice` | Return the Tamari lattice. 

:meth:`~posets.TetrahedralPoset` | Return the Tetrahedral poset with `n-1` layers based on the input colors. 

:meth:`~posets.UpDownPoset` | Return the up-down poset on `n` elements. 

:meth:`~posets.YoungDiagramPoset` | Return the poset of cells in the Young diagram of a partition. 

:meth:`~posets.YoungsLattice` | Return Young's Lattice up to rank `n`. 

:meth:`~posets.YoungsLatticePrincipalOrderIdeal` | Return the principal order ideal of the partition `lam` in Young's Lattice. 

 

Constructions 

------------- 

""" 

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

# Copyright (C) 2008 Peter Jipsen <jipsen@chapman.edu>, 

# Franco Saliola <saliola@gmail.com> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# 

# This code is distributed in the hope that it will be useful, 

# but WITHOUT ANY WARRANTY; without even the implied warranty of 

# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 

# General Public License for more details. 

# 

# The full text of the GPL is available at: 

# 

# http://www.gnu.org/licenses/ 

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

from __future__ import print_function 

from six import add_metaclass, string_types 

 

from sage.misc.classcall_metaclass import ClasscallMetaclass 

import sage.categories.posets 

from sage.combinat.permutation import Permutations, Permutation 

from sage.combinat.posets.posets import Poset, FinitePoset, FinitePosets_n 

from sage.combinat.posets.lattices import (LatticePoset, MeetSemilattice, 

JoinSemilattice, FiniteLatticePoset) 

from sage.categories.finite_posets import FinitePosets 

from sage.categories.finite_lattice_posets import FiniteLatticePosets 

from sage.graphs.digraph import DiGraph 

from sage.rings.integer import Integer 

 

 

@add_metaclass(ClasscallMetaclass) 

class Posets(object): 

r""" 

A collection of posets and lattices. 

 

EXAMPLES:: 

 

sage: posets.BooleanLattice(3) 

Finite lattice containing 8 elements 

sage: posets.ChainPoset(3) 

Finite lattice containing 3 elements 

sage: posets.RandomPoset(17,.15) 

Finite poset containing 17 elements 

 

The category of all posets:: 

 

sage: Posets() 

Category of posets 

 

The enumerated set of all posets on `3` elements, up to an 

isomorphism:: 

 

sage: Posets(3) 

Posets containing 3 elements 

 

.. SEEALSO:: :class:`~sage.categories.posets.Posets`, :class:`FinitePosets`, :func:`Poset` 

 

TESTS:: 

 

sage: P = Posets 

sage: TestSuite(P).run() 

""" 

@staticmethod 

def __classcall__(cls, n = None): 

r""" 

Return either the category of all posets, or the finite 

enumerated set of all finite posets on ``n`` elements up to an 

isomorphism. 

 

EXAMPLES:: 

 

sage: Posets() 

Category of posets 

sage: Posets(4) 

Posets containing 4 elements 

""" 

if n is None: 

return sage.categories.posets.Posets() 

try: 

n = Integer(n) 

except TypeError: 

raise TypeError("number of elements must be an integer, not {0}".format(n)) 

if n < 0: 

raise ValueError("number of elements must be non-negative, not {0}".format(n)) 

return FinitePosets_n(n) 

 

@staticmethod 

def BooleanLattice(n, facade=None): 

""" 

Return the Boolean lattice containing `2^n` elements. 

 

- ``n`` (an integer) -- number of elements will be `2^n` 

- ``facade`` (boolean) -- whether to make the returned poset a 

facade poset (see :mod:`sage.categories.facade_sets`); the 

default behaviour is the same as the default behaviour of 

the :func:`~sage.combinat.posets.posets.Poset` constructor 

 

EXAMPLES:: 

 

sage: posets.BooleanLattice(5) 

Finite lattice containing 32 elements 

""" 

try: 

n = Integer(n) 

except TypeError: 

raise TypeError("number of elements must be an integer, not {0}".format(n)) 

if n < 0: 

raise ValueError("number of elements must be non-negative, not {0}".format(n)) 

if n == 0: 

return LatticePoset( ([0], []) ) 

if n == 1: 

return LatticePoset( ([0,1], [[0,1]]) ) 

L = [[Integer(x|(1<<y)) for y in range(n) if x&(1<<y)==0] for 

x in range(2**n)] 

D = DiGraph({v: L[v] for v in range(2**n)}) 

return FiniteLatticePoset(hasse_diagram=D, 

category=FiniteLatticePosets(), 

facade=facade) 

 

@staticmethod 

def ChainPoset(n, facade=None): 

""" 

Return a chain (a totally ordered poset) containing ``n`` elements. 

 

- ``n`` (an integer) -- number of elements. 

- ``facade`` (boolean) -- whether to make the returned poset a 

facade poset (see :mod:`sage.categories.facade_sets`); the 

default behaviour is the same as the default behaviour of 

the :func:`~sage.combinat.posets.posets.Poset` constructor 

 

EXAMPLES:: 

 

sage: C = posets.ChainPoset(6); C 

Finite lattice containing 6 elements 

sage: C.linear_extension() 

[0, 1, 2, 3, 4, 5] 

 

TESTS:: 

 

sage: for i in range(5): 

....: for j in range(5): 

....: if C.covers(C(i),C(j)) and j != i+1: 

....: print("TEST FAILED") 

 

Check that :trac:`8422` is solved:: 

 

sage: posets.ChainPoset(0) 

Finite lattice containing 0 elements 

sage: C = posets.ChainPoset(1); C 

Finite lattice containing 1 elements 

sage: C.cover_relations() 

[] 

sage: C = posets.ChainPoset(2); C 

Finite lattice containing 2 elements 

sage: C.cover_relations() 

[[0, 1]] 

""" 

try: 

n = Integer(n) 

except TypeError: 

raise TypeError("number of elements must be an integer, not {0}".format(n)) 

if n < 0: 

raise ValueError("number of elements must be non-negative, not {0}".format(n)) 

D = DiGraph([range(n), [[x,x+1] for x in range(n-1)]], 

format='vertices_and_edges') 

return FiniteLatticePoset(hasse_diagram=D, 

category=FiniteLatticePosets(), 

facade=facade) 

 

@staticmethod 

def AntichainPoset(n, facade=None): 

""" 

Return an antichain (a poset with no comparable elements) 

containing `n` elements. 

 

INPUT: 

 

- ``n`` (an integer) -- number of elements 

- ``facade`` (boolean) -- whether to make the returned poset a 

facade poset (see :mod:`sage.categories.facade_sets`); the 

default behaviour is the same as the default behaviour of 

the :func:`~sage.combinat.posets.posets.Poset` constructor 

 

EXAMPLES:: 

 

sage: A = posets.AntichainPoset(6); A 

Finite poset containing 6 elements 

 

TESTS:: 

 

sage: for i in range(5): 

....: for j in range(5): 

....: if A.covers(A(i),A(j)): 

....: print("TEST FAILED") 

 

TESTS: 

 

Check that :trac:`8422` is solved:: 

 

sage: posets.AntichainPoset(0) 

Finite poset containing 0 elements 

sage: C = posets.AntichainPoset(1); C 

Finite poset containing 1 elements 

sage: C.cover_relations() 

[] 

sage: C = posets.AntichainPoset(2); C 

Finite poset containing 2 elements 

sage: C.cover_relations() 

[] 

""" 

try: 

n = Integer(n) 

except TypeError: 

raise TypeError("number of elements must be an integer, not {0}".format(n)) 

if n < 0: 

raise ValueError("number of elements must be non-negative, not {0}".format(n)) 

return Poset((range(n), []), facade=facade) 

 

@staticmethod 

def PentagonPoset(facade=None): 

""" 

Return the Pentagon poset. 

 

INPUT: 

 

- ``facade`` (boolean) -- whether to make the returned poset a 

facade poset (see :mod:`sage.categories.facade_sets`); the 

default behaviour is the same as the default behaviour of 

the :func:`~sage.combinat.posets.posets.Poset` constructor 

 

EXAMPLES:: 

 

sage: P = posets.PentagonPoset(); P 

Finite lattice containing 5 elements 

sage: P.cover_relations() 

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

 

TESTS: 

 

This is smallest lattice that is not modular:: 

 

sage: P.is_modular() 

False 

 

This poset and the :meth:`DiamondPoset` are the two smallest 

lattices which are not distributive:: 

 

sage: P.is_distributive() 

False 

sage: posets.DiamondPoset(5).is_distributive() 

False 

""" 

return LatticePoset([[1,2],[4],[3],[4],[]], facade=facade) 

 

@staticmethod 

def DiamondPoset(n, facade=None): 

""" 

Return the lattice of rank two containing ``n`` elements. 

 

INPUT: 

 

- ``n`` -- number of elements, an integer at least 3 

 

- ``facade`` (boolean) -- whether to make the returned poset a 

facade poset (see :mod:`sage.categories.facade_sets`); the 

default behaviour is the same as the default behaviour of 

the :func:`~sage.combinat.posets.posets.Poset` constructor 

 

EXAMPLES:: 

 

sage: posets.DiamondPoset(7) 

Finite lattice containing 7 elements 

""" 

try: 

n = Integer(n) 

except TypeError: 

raise TypeError("number of elements must be an integer, not {0}".format(n)) 

if n <= 2: 

raise ValueError("n must be an integer at least 3") 

c = [[n-1] for x in range(n)] 

c[0] = [x for x in range(1,n-1)] 

c[n-1] = [] 

D = DiGraph({v:c[v] for v in range(n)}, format='dict_of_lists') 

return FiniteLatticePoset(hasse_diagram=D, 

category=FiniteLatticePosets(), 

facade=facade) 

 

@staticmethod 

def Crown(n, facade=None): 

""" 

Return the crown poset of `2n` elements. 

 

In this poset every element `i` for `0 \leq i \leq n-1` 

is covered by elements `i+n` and `i+n+1`, except that 

`n-1` is covered by `n` and `n+1`. 

 

INPUT: 

 

- ``n`` -- number of elements, an integer at least 2 

 

- ``facade`` (boolean) -- whether to make the returned poset a 

facade poset (see :mod:`sage.categories.facade_sets`); the 

default behaviour is the same as the default behaviour of 

the :func:`~sage.combinat.posets.posets.Poset` constructor 

 

EXAMPLES:: 

 

sage: posets.Crown(3) 

Finite poset containing 6 elements 

""" 

try: 

n = Integer(n) 

except TypeError: 

raise TypeError("number of elements must be an integer, not {0}".format(n)) 

if n < 2: 

raise ValueError("n must be an integer at least 2") 

D = {i: [i+n, i+n+1] for i in range(n-1)} 

D[n-1] = [n, n+n-1] 

return FinitePoset(hasse_diagram=DiGraph(D), category=FinitePosets(), 

facade=facade) 

 

@staticmethod 

def DivisorLattice(n, facade=None): 

""" 

Return the divisor lattice of an integer. 

 

Elements of the lattice are divisors of `n` and `x < y` in the 

lattice if `x` divides `y`. 

 

INPUT: 

 

- ``n`` -- an integer 

- ``facade`` (boolean) -- whether to make the returned poset a 

facade poset (see :mod:`sage.categories.facade_sets`); the 

default behaviour is the same as the default behaviour of 

the :func:`~sage.combinat.posets.posets.Poset` constructor 

 

EXAMPLES:: 

 

sage: P = posets.DivisorLattice(12) 

sage: sorted(P.cover_relations()) 

[[1, 2], [1, 3], [2, 4], [2, 6], [3, 6], [4, 12], [6, 12]] 

 

sage: P = posets.DivisorLattice(10, facade=False) 

sage: P(2) < P(5) 

False 

 

TESTS:: 

 

sage: posets.DivisorLattice(1) 

Finite lattice containing 1 elements with distinguished linear extension 

""" 

from sage.arith.misc import divisors, is_prime 

try: 

n = Integer(n) 

except TypeError: 

raise TypeError("number of elements must be an integer, not {0}".format(n)) 

if n <= 0: 

raise ValueError("n must be a positive integer") 

Div_n = divisors(n) 

hasse = DiGraph([Div_n, lambda a, b: b%a==0 and is_prime(b//a)]) 

return FiniteLatticePoset(hasse, elements=Div_n, facade=facade, 

category=FiniteLatticePosets()) 

 

@staticmethod 

def IntegerCompositions(n): 

""" 

Return the poset of integer compositions of the integer ``n``. 

 

A composition of a positive integer `n` is a list of positive 

integers that sum to `n`. The order is reverse refinement: 

`[p_1,p_2,...,p_l] < [q_1,q_2,...,q_m]` if `q` consists 

of an integer composition of `p_1`, followed by an integer 

composition of `p_2`, and so on. 

 

EXAMPLES:: 

 

sage: P = posets.IntegerCompositions(7); P 

Finite poset containing 64 elements 

sage: len(P.cover_relations()) 

192 

""" 

from sage.combinat.composition import Compositions 

C = Compositions(n) 

return Poset((C, [[c,d] for c in C for d in C if d.is_finer(c)]), cover_relations=False) 

 

@staticmethod 

def IntegerPartitions(n): 

""" 

Return the poset of integer partitions on the integer ``n``. 

 

A partition of a positive integer `n` is a non-increasing list 

of positive integers that sum to `n`. If `p` and `q` are 

integer partitions of `n`, then `p` covers `q` if and only 

if `q` is obtained from `p` by joining two parts of `p` 

(and sorting, if necessary). 

 

EXAMPLES:: 

 

sage: P = posets.IntegerPartitions(7); P 

Finite poset containing 15 elements 

sage: len(P.cover_relations()) 

28 

""" 

def lower_covers(partition): 

r""" 

Nested function for computing the lower covers 

of elements in the poset of integer partitions. 

""" 

lc = [] 

for i in range(len(partition)-1): 

for j in range(i+1,len(partition)): 

new_partition = partition[:] 

del new_partition[j] 

del new_partition[i] 

new_partition.append(partition[i]+partition[j]) 

new_partition.sort(reverse=True) 

tup = tuple(new_partition) 

if tup not in lc: 

lc.append(tup) 

return lc 

from sage.combinat.partition import Partitions 

H = DiGraph(dict([[tuple(p),lower_covers(p)] for p in Partitions(n)])) 

return Poset(H.reverse()) 

 

@staticmethod 

def RestrictedIntegerPartitions(n): 

""" 

Return the poset of integer partitions on the integer `n` 

ordered by restricted refinement. 

 

That is, if `p` and `q` are integer partitions of `n`, then 

`p` covers `q` if and only if `q` is obtained from `p` by 

joining two distinct parts of `p` (and sorting, if necessary). 

 

EXAMPLES:: 

 

sage: P = posets.RestrictedIntegerPartitions(7); P 

Finite poset containing 15 elements 

sage: len(P.cover_relations()) 

17 

 

""" 

def lower_covers(partition): 

r""" 

Nested function for computing the lower covers of elements in the 

restricted poset of integer partitions. 

""" 

lc = [] 

for i in range(len(partition)-1): 

for j in range(i+1,len(partition)): 

if partition[i] != partition[j]: 

new_partition = partition[:] 

del new_partition[j] 

del new_partition[i] 

new_partition.append(partition[i]+partition[j]) 

new_partition.sort(reverse=True) 

tup = tuple(new_partition) 

if tup not in lc: 

lc.append(tup) 

return lc 

from sage.combinat.partition import Partitions 

H = DiGraph(dict([[tuple(p),lower_covers(p)] for p in Partitions(n)])) 

return Poset(H.reverse()) 

 

@staticmethod 

def IntegerPartitionsDominanceOrder(n): 

r""" 

Return the lattice of integer partitions on the integer `n` 

ordered by dominance. 

 

That is, if `p=(p_1,\ldots,p_i)` and `q=(q_1,\ldots,q_j)` are 

integer partitions of `n`, then `p` is greater than `q` if and 

only if `p_1+\cdots+p_k > q_1+\cdots+q_k` for all `k`. 

 

INPUT: 

 

- ``n`` -- a positive integer 

 

EXAMPLES:: 

 

sage: P = posets.IntegerPartitionsDominanceOrder(6); P 

Finite lattice containing 11 elements 

sage: P.cover_relations() 

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

[[2, 1, 1, 1, 1], [2, 2, 1, 1]], 

[[2, 2, 1, 1], [2, 2, 2]], 

[[2, 2, 1, 1], [3, 1, 1, 1]], 

[[2, 2, 2], [3, 2, 1]], 

[[3, 1, 1, 1], [3, 2, 1]], 

[[3, 2, 1], [3, 3]], 

[[3, 2, 1], [4, 1, 1]], 

[[3, 3], [4, 2]], 

[[4, 1, 1], [4, 2]], 

[[4, 2], [5, 1]], 

[[5, 1], [6]]] 

""" 

from sage.rings.semirings.non_negative_integer_semiring import NN 

if n not in NN: 

raise ValueError('n must be an integer') 

from sage.combinat.partition import Partitions, Partition 

return LatticePoset((Partitions(n), Partition.dominates)).dual() 

 

@staticmethod 

def PowerPoset(n): 

""" 

Return the power poset on `n` element posets. 

 

Elements of the power poset are all posets on 

the set `\{0, 1, \ldots, n-1\}` ordered by extension. 

That is, the antichain of `n` elements is the bottom and 

`P_a \le P_b` in the power poset if `P_b` is an extension 

of `P_a`. 

 

These were studied in [Bru1994]_. 

 

EXAMPLES:: 

 

sage: P3 = posets.PowerPoset(3); P3 

Finite meet-semilattice containing 19 elements 

sage: all(P.is_chain() for P in P3.maximal_elements()) 

True 

 

TESTS:: 

 

sage: P0 = posets.PowerPoset(0); P0 

Finite meet-semilattice containing 1 elements 

sage: P0[0] 

Finite poset containing 0 elements 

sage: P1 = posets.PowerPoset(1); P1 

Finite meet-semilattice containing 1 elements 

sage: P1[0] 

Finite poset containing 1 elements 

sage: P1[0][0] 

0 

""" 

# Todo: Make this faster. 

 

try: 

n = Integer(n) 

except TypeError: 

raise TypeError("parameter n must be an integer, not {0}".format(n)) 

if n < 0: 

raise ValueError("parameter n must be non-negative, not {0}".format(n)) 

 

all_pos_n = set() 

Pn = list(Posets(n)) 

for P in Pn: 

for r in Permutations(P): 

all_pos_n.add(P.relabel(list(r))) 

 

return MeetSemilattice((all_pos_n, 

lambda A, B: all(B.is_lequal(x, y) for x,y in A.cover_relations_iterator()) 

)) 

 

@staticmethod 

def RandomPoset(n, p): 

r""" 

Generate a random poset on ``n`` elements according to a 

probability ``p``. 

 

INPUT: 

 

- ``n`` - number of elements, a non-negative integer 

 

- ``p`` - a probability, a real number between 0 and 1 (inclusive) 

 

OUTPUT: 

 

A poset on `n` elements. The probability `p` roughly measures 

width/height of the output: `p=0` always generates an antichain, 

`p=1` will return a chain. To create interesting examples, 

keep the probability small, perhaps on the order of `1/n`. 

 

EXAMPLES:: 

 

sage: set_random_seed(0) # Results are reproducible 

sage: P = posets.RandomPoset(5, 0.3) 

sage: P.cover_relations() 

[[5, 4], [4, 2], [1, 2]] 

 

.. SEEALSO:: :meth:`RandomLattice` 

 

TESTS:: 

 

sage: posets.RandomPoset('junk', 0.5) 

Traceback (most recent call last): 

... 

TypeError: number of elements must be an integer, not junk 

 

sage: posets.RandomPoset(-6, 0.5) 

Traceback (most recent call last): 

... 

ValueError: number of elements must be non-negative, not -6 

 

sage: posets.RandomPoset(6, 'garbage') 

Traceback (most recent call last): 

... 

TypeError: probability must be a real number, not garbage 

 

sage: posets.RandomPoset(6, -0.5) 

Traceback (most recent call last): 

... 

ValueError: probability must be between 0 and 1, not -0.5 

 

sage: posets.RandomPoset(0, 0.5) 

Finite poset containing 0 elements 

""" 

from sage.misc.prandom import random 

 

try: 

n = Integer(n) 

except TypeError: 

raise TypeError("number of elements must be an integer, not {0}".format(n)) 

if n < 0: 

raise ValueError("number of elements must be non-negative, not {0}".format(n)) 

try: 

p = float(p) 

except Exception: 

raise TypeError("probability must be a real number, not {0}".format(p)) 

if p < 0 or p> 1: 

raise ValueError("probability must be between 0 and 1, not {0}".format(p)) 

 

D = DiGraph(loops=False, multiedges=False) 

D.add_vertices(range(n)) 

for i in range(n): 

for j in range(i+1, n): 

if random() < p: 

D.add_edge(i, j) 

D.relabel(list(Permutations(n).random_element())) 

return Poset(D, cover_relations=False) 

 

@staticmethod 

def RandomLattice(n, p, properties=None): 

r""" 

Return a random lattice on ``n`` elements. 

 

INPUT: 

 

- ``n`` -- number of elements, a non-negative integer 

 

- ``p`` -- a probability, a positive real number less than one 

 

- ``properties`` -- a list of properties for the lattice. Currently 

implemented: 

 

* ``None``, no restrictions for lattices to create 

* ``'planar'``, the lattice has an upward planar drawing 

* ``'dismantlable'`` (implicated by ``'planar'``) 

* ``'distributive'`` (implicated by ``'stone'``) 

* ``'stone'`` 

 

OUTPUT: 

 

A lattice on `n` elements. When ``properties`` is ``None``, 

the probability `p` roughly measures number of covering 

relations of the lattice. To create interesting examples, make 

the probability near one, something like `0.98..0.999`. 

 

Currently parameter ``p`` has no effect only when ``properties`` 

is not ``None``. 

 

.. NOTE:: 

 

Results are reproducible in same Sage version only. Underlying 

algorithm may change in future versions. 

 

EXAMPLES:: 

 

sage: set_random_seed(0) # Results are reproducible 

sage: L = posets.RandomLattice(8, 0.995); L 

Finite lattice containing 8 elements 

sage: L.cover_relations() 

[[7, 6], [7, 3], [7, 1], ..., [5, 4], [2, 4], [1, 4], [0, 4]] 

sage: L = posets.RandomLattice(10, 0, properties=['dismantlable']) 

sage: L.is_dismantlable() 

True 

 

.. SEEALSO:: :meth:`RandomPoset` 

 

TESTS:: 

 

sage: posets.RandomLattice('junk', 0.5) 

Traceback (most recent call last): 

... 

TypeError: number of elements must be an integer, not junk 

 

sage: posets.RandomLattice(-6, 0.5) 

Traceback (most recent call last): 

... 

ValueError: number of elements must be non-negative, not -6 

 

sage: posets.RandomLattice(6, 'garbage') 

Traceback (most recent call last): 

... 

TypeError: probability must be a real number, not garbage 

 

sage: posets.RandomLattice(6, -0.5) 

Traceback (most recent call last): 

... 

ValueError: probability must be a positive real number and below 1, not -0.5 

 

sage: posets.RandomLattice(10, 0.5, properties=['junk']) 

Traceback (most recent call last): 

... 

ValueError: unknown value junk for 'properties' 

 

sage: posets.RandomLattice(0, 0.5) 

Finite lattice containing 0 elements 

""" 

from copy import copy 

 

try: 

n = Integer(n) 

except TypeError: 

raise TypeError("number of elements must be an integer, not {0}".format(n)) 

if n < 0: 

raise ValueError("number of elements must be non-negative, not {0}".format(n)) 

try: 

p = float(p) 

except Exception: 

raise TypeError("probability must be a real number, not {0}".format(p)) 

if p < 0 or p >= 1: 

raise ValueError("probability must be a positive real number and below 1, not {0}".format(p)) 

 

if properties is None: 

# Basic case, no special properties for lattice asked. 

if n <= 3: 

return posets.ChainPoset(n) 

covers = _random_lattice(n, p) 

covers_dict = {i:covers[i] for i in range(n)} 

D = DiGraph(covers_dict) 

D.relabel([i-1 for i in Permutations(n).random_element()]) 

return LatticePoset(D, cover_relations=True) 

 

if isinstance(properties, string_types): 

properties = set([properties]) 

else: 

properties = set(properties) 

 

known_properties = set(['planar', 'dismantlable', 'distributive', 'stone']) 

errors = properties.difference(known_properties) 

if errors: 

raise ValueError("unknown value %s for 'properties'" % errors.pop()) 

 

if n <= 3: 

# Change this, if property='complemented' is added 

return posets.ChainPoset(n) 

 

# Handling properties: planar => dismantlable, stone => distributive 

if 'planar' in properties: 

properties.discard('dismantlable') 

if 'stone' in properties: 

properties.discard('distributive') 

 

# Test property combinations that are not implemented. 

if 'distributive' in properties and len(properties) > 1: 

raise NotImplementedError("combining 'distributive' with other properties is not implemented") 

if 'stone' in properties and len(properties) > 1: 

raise NotImplementedError("combining 'stone' with other properties is not implemented") 

 

if properties == set(['planar']): 

D = _random_planar_lattice(n) 

D.relabel([i-1 for i in Permutations(n).random_element()]) 

return LatticePoset(D) 

 

if properties == set(['dismantlable']): 

D = _random_dismantlable_lattice(n) 

D.relabel([i-1 for i in Permutations(n).random_element()]) 

return LatticePoset(D) 

 

if properties == set(['stone']): 

D = _random_stone_lattice(n) 

D.relabel([i-1 for i in Permutations(n).random_element()]) 

return LatticePoset(D) 

 

if properties == set(['distributive']): 

tmp = Poset(_random_distributive_lattice(n)).order_ideals_lattice(as_ideals=False) 

D = copy(tmp._hasse_diagram) 

D.relabel([i-1 for i in Permutations(n).random_element()]) 

return LatticePoset(D) 

 

raise AssertionError("Bug in RandomLattice().") 

 

@staticmethod 

def SetPartitions(n): 

r""" 

Return the lattice of set partitions of the set `\{1,\ldots,n\}` 

ordered by refinement. 

 

INPUT: 

 

- ``n`` -- a positive integer 

 

EXAMPLES:: 

 

sage: posets.SetPartitions(4) 

Finite lattice containing 15 elements 

""" 

from sage.rings.semirings.non_negative_integer_semiring import NN 

if n not in NN: 

raise ValueError('n must be an integer') 

from sage.combinat.set_partition import SetPartitions 

S = SetPartitions(n) 

 

def covers(x): 

for i, s in enumerate(x): 

for j in range(i+1, len(x)): 

L = list(x) 

L[i] = s.union(x[j]) 

L.pop(j) 

yield S(L) 

 

return LatticePoset({x: list(covers(x)) for x in S}, 

cover_relations=True) 

 

@staticmethod 

def SSTPoset(s, f=None): 

""" 

The poset on semistandard tableaux of shape ``s`` and largest 

entry ``f`` that is ordered by componentwise comparison of the 

entries. 

 

INPUT: 

 

- ``s`` - shape of the tableaux 

 

- ``f`` - maximum fill number. This is an optional 

argument. If no maximal number is given, it will use 

the number of cells in the shape. 

 

NOTE: This is a basic implementation and most certainly 

not the most efficient. 

 

EXAMPLES:: 

 

sage: posets.SSTPoset([2,1]) 

Finite poset containing 8 elements 

 

sage: posets.SSTPoset([2,1],4) 

Finite poset containing 20 elements 

 

sage: posets.SSTPoset([2,1],2).cover_relations() 

[[[[1, 1], [2]], [[1, 2], [2]]]] 

 

sage: posets.SSTPoset([3,2]).bottom() # long time (6s on sage.math, 2012) 

[[1, 1, 1], [2, 2]] 

 

sage: posets.SSTPoset([3,2],4).maximal_elements() 

[[[3, 3, 4], [4, 4]]] 

""" 

from sage.combinat.tableau import SemistandardTableaux 

 

def tableaux_is_less_than(a, b): 

atstring = [] 

btstring = [] 

for i in a: 

atstring += i 

for i in b: 

btstring += i 

for i in range(len(atstring)): 

if atstring[i] > btstring[i]: 

return False 

return True 

if f is None: 

f=0 

for i in s: 

f += i 

E = SemistandardTableaux(s, max_entry=f) 

return Poset((E, tableaux_is_less_than)) 

 

@staticmethod 

def StandardExample(n, facade=None): 

r""" 

Return the partially ordered set on ``2n`` elements with 

dimension ``n``. 

 

Let `P` be the poset on `\{0, 1, 2, \ldots, 2n-1\}` whose defining 

relations are that `i < j` for every `0 \leq i < n \leq j < 2n` 

except when `i + n = j`. The poset `P` is the so-called 

*standard example* of a poset with dimension `n`. 

 

INPUT: 

 

- ``n`` -- an integer `\ge 2`, dimension of the constructed poset 

- ``facade`` (boolean) -- whether to make the returned poset a 

facade poset (see :mod:`sage.categories.facade_sets`); the 

default behaviour is the same as the default behaviour of 

the :func:`~sage.combinat.posets.posets.Poset` constructor 

 

OUTPUT: 

 

The standard example of a poset of dimension `n`. 

 

EXAMPLES:: 

 

sage: A = posets.StandardExample(3); A 

Finite poset containing 6 elements 

sage: A.dimension() 

3 

 

REFERENCES: 

 

- [Gar2015]_ 

- [Ros1999]_ 

 

TESTS:: 

 

sage: A = posets.StandardExample(10); A 

Finite poset containing 20 elements 

sage: len(A.cover_relations()) 

90 

 

sage: P = posets.StandardExample(5, facade=False) 

sage: P(4) < P(3), P(4) > P(3) 

(False, False) 

""" 

try: 

n = Integer(n) 

except TypeError: 

raise TypeError("dimension must be an integer, not {0}".format(n)) 

if n < 2: 

raise ValueError("dimension must be at least 2, not {0}".format(n)) 

return Poset((range(2*n), [[i, j+n] for i in range(n) 

for j in range(n) if i != j]), 

facade=facade) 

 

@staticmethod 

def SymmetricGroupBruhatOrderPoset(n): 

""" 

The poset of permutations with respect to Bruhat order. 

 

EXAMPLES:: 

 

sage: posets.SymmetricGroupBruhatOrderPoset(4) 

Finite poset containing 24 elements 

""" 

if n < 10: 

element_labels = {s: "".join(str(x) for x in s) 

for s in Permutations(n)} 

return Poset({s: s.bruhat_succ() for s in Permutations(n)}, 

element_labels) 

 

@staticmethod 

def SymmetricGroupBruhatIntervalPoset(start, end): 

""" 

The poset of permutations with respect to Bruhat order. 

 

INPUT: 

 

- ``start`` - list permutation 

 

- ``end`` - list permutation (same n, of course) 

 

.. note:: 

 

Must have ``start`` <= ``end``. 

 

EXAMPLES: 

 

Any interval is rank symmetric if and only if it avoids these 

permutations:: 

 

sage: P1 = posets.SymmetricGroupBruhatIntervalPoset([1,2,3,4], [3,4,1,2]) 

sage: P2 = posets.SymmetricGroupBruhatIntervalPoset([1,2,3,4], [4,2,3,1]) 

sage: ranks1 = [P1.rank(v) for v in P1] 

sage: ranks2 = [P2.rank(v) for v in P2] 

sage: [ranks1.count(i) for i in uniq(ranks1)] 

[1, 3, 5, 4, 1] 

sage: [ranks2.count(i) for i in uniq(ranks2)] 

[1, 3, 5, 6, 4, 1] 

 

""" 

start = Permutation(start) 

end = Permutation(end) 

if len(start) != len(end): 

raise TypeError("Start (%s) and end (%s) must have same length." % (start, end)) 

if not start.bruhat_lequal(end): 

raise TypeError("Must have start (%s) <= end (%s) in Bruhat order." % (start, end)) 

unseen = [start] 

nodes = {} 

while unseen: 

perm = unseen.pop(0) 

nodes[perm] = [succ_perm for succ_perm in perm.bruhat_succ() 

if succ_perm.bruhat_lequal(end)] 

for succ_perm in nodes[perm]: 

if succ_perm not in nodes: 

unseen.append(succ_perm) 

return Poset(nodes) 

 

@staticmethod 

def SymmetricGroupWeakOrderPoset(n, labels="permutations", side="right"): 

r""" 

The poset of permutations of `\{ 1, 2, \ldots, n \}` with respect 

to the weak order (also known as the permutohedron order, cf. 

:meth:`~sage.combinat.permutation.Permutation.permutohedron_lequal`). 

 

The optional variable ``labels`` (default: ``"permutations"``) 

determines the labelling of the elements if `n < 10`. The optional 

variable ``side`` (default: ``"right"``) determines whether the 

right or the left permutohedron order is to be used. 

 

EXAMPLES:: 

 

sage: posets.SymmetricGroupWeakOrderPoset(4) 

Finite poset containing 24 elements 

""" 

if n < 10 and labels == "permutations": 

element_labels = dict([[s,"".join(map(str,s))] for s in Permutations(n)]) 

if n < 10 and labels == "reduced_words": 

element_labels = dict([[s,"".join(map(str,s.reduced_word_lexmin()))] for s in Permutations(n)]) 

if side == "left": 

 

def weak_covers(s): 

r""" 

Nested function for computing the covers of elements in the 

poset of left weak order for the symmetric group. 

""" 

return [v for v in s.bruhat_succ() if 

s.length() + (s.inverse().right_action_product(v)).length() == v.length()] 

else: 

def weak_covers(s): 

r""" 

Nested function for computing the covers of elements in the 

poset of right weak order for the symmetric group. 

""" 

return [v for v in s.bruhat_succ() if 

s.length() + (s.inverse().left_action_product(v)).length() == v.length()] 

return Poset(dict([[s, weak_covers(s)] for s in Permutations(n)]),element_labels) 

 

@staticmethod 

def TetrahedralPoset(n, *colors, **labels): 

r""" 

Return the tetrahedral poset based on the input colors. 

 

This method will return the tetrahedral poset with n-1 layers and 

covering relations based on the input colors of 'green', 'red', 

'orange', 'silver', 'yellow' and 'blue' as defined in [Striker2011]_. 

For particular color choices, the order ideals of the resulting 

tetrahedral poset will be isomorphic to known combinatorial objects. 

 

For example, for the colors 'blue', 'yellow', 'orange', and 'green', 

the order ideals will be in bijection with alternating sign matrices. 

For the colors 'yellow', 'orange', and 'green', the order ideals will 

be in bijection with semistandard Young tableaux of staircase shape. 

For the colors 'red', 'orange', 'green', and optionally 'yellow', the 

order ideals will be in bijection with totally symmetric 

self-complementary plane partitions in a `2n \times 2n \times 2n` box. 

 

INPUT: 

 

- ``n`` - Defines the number (n-1) of layers in the poset. 

 

- ``colors`` - The colors that define the covering relations of the 

poset. Colors used are 'green', 'red', 'yellow', 'orange', 'silver', 

and 'blue'. 

 

- ``labels`` - Keyword variable used to determine whether the poset 

is labeled with integers or tuples. To label with integers, the 

method should be called with ``labels='integers'``. Otherwise, the 

labeling will default to tuples. 

 

EXAMPLES:: 

 

sage: posets.TetrahedralPoset(4,'green','red','yellow','silver','blue','orange') 

Finite poset containing 10 elements 

 

sage: posets.TetrahedralPoset(4,'green','red','yellow','silver','blue','orange', labels='integers') 

Finite poset containing 10 elements 

 

sage: A = AlternatingSignMatrices(3) 

sage: p = A.lattice() 

sage: ji = p.join_irreducibles_poset() 

sage: tet = posets.TetrahedralPoset(3, 'green','yellow','blue','orange') 

sage: ji.is_isomorphic(tet) 

True 

""" 

n = n - 1 

try: 

n = Integer(n) 

except TypeError: 

raise TypeError("n must be an integer.") 

if n < 2: 

raise ValueError("n must be greater than 2.") 

for c in colors: 

if(c not in ('green', 'red', 'yellow', 'orange', 'silver', 'blue')): 

raise ValueError("Color input must be from the following: 'green', 'red', 'yellow', 'orange', 'silver', and 'blue'.") 

elem=[(i,j,k) for i in range (n) for j in range (n-i) for k in range (n-i-j)] 

rels = [] 

elem_labels = {} 

if 'labels' in labels: 

if labels['labels'] == 'integers': 

labelcount = 0 

for (i,j,k) in elem: 

elem_labels[(i,j,k)] = labelcount 

labelcount += 1 

for c in colors: 

for (i,j,k) in elem: 

if(i+j+k < n-1): 

if(c=='green'): 

rels.append([(i,j,k),(i+1,j,k)]) 

if(c=='red'): 

rels.append([(i,j,k),(i,j,k+1)]) 

if(c=='yellow'): 

rels.append([(i,j,k),(i,j+1,k)]) 

if(j<n-1 and k>0): 

if(c=='orange'): 

rels.append([(i,j,k),(i,j+1,k-1)]) 

if(i<n-1 and j>0): 

if(c=='silver'): 

rels.append([(i,j,k),(i+1,j-1,k)]) 

if(i<n-1 and k>0): 

if(c=='blue'): 

rels.append([(i,j,k),(i+1,j,k-1)]) 

return Poset([elem,rels], elem_labels) 

 

# shard intersection order 

import sage.combinat.shard_order 

ShardPoset = staticmethod(sage.combinat.shard_order.shard_poset) 

 

# Tamari lattices 

import sage.combinat.tamari_lattices 

TamariLattice = staticmethod(sage.combinat.tamari_lattices.TamariLattice) 

 

@staticmethod 

def CoxeterGroupAbsoluteOrderPoset(W, use_reduced_words=True): 

r""" 

Return the poset of elements of a Coxeter group with respect 

to absolute order. 

 

INPUT: 

 

- ``W`` -- a Coxeter group 

- ``use_reduced_words`` -- boolean (default: ``True``); if 

``True``, then the elements are labeled by their lexicographically 

minimal reduced word 

 

EXAMPLES:: 

 

sage: W = CoxeterGroup(['B', 3]) 

sage: posets.CoxeterGroupAbsoluteOrderPoset(W) 

Finite poset containing 48 elements 

 

sage: W = WeylGroup(['B', 2], prefix='s') 

sage: posets.CoxeterGroupAbsoluteOrderPoset(W, False) 

Finite poset containing 8 elements 

""" 

if use_reduced_words: 

element_labels = {s: tuple(s.reduced_word()) for s in W} 

return Poset({s: s.absolute_covers() for s in W}, element_labels) 

return Poset({s: s.absolute_covers() for s in W}) 

 

@staticmethod 

def NoncrossingPartitions(W): 

""" 

Return the lattice of noncrossing partitions. 

 

INPUT: 

 

- ``W`` -- a finite Coxeter group or a Weyl group 

 

EXAMPLES:: 

 

sage: W = CoxeterGroup(['A', 3]) 

sage: posets.NoncrossingPartitions(W) 

Finite lattice containing 14 elements 

 

sage: W = WeylGroup(['B', 2], prefix='s') 

sage: posets.NoncrossingPartitions(W) 

Finite lattice containing 6 elements 

""" 

return W.noncrossing_partition_lattice() 

 

@staticmethod 

def SymmetricGroupAbsoluteOrderPoset(n, labels="permutations"): 

r""" 

Return the poset of permutations with respect to absolute order. 

 

INPUT: 

 

- ``n`` -- a positive integer 

 

- ``label`` -- (default: ``'permutations'``) a label for the elements 

of the poset returned by the function; the options are 

 

* ``'permutations'`` - labels the elements are given by their 

one-line notation 

* ``'reduced_words'`` - labels the elements by the 

lexicographically minimal reduced word 

* ``'cycles'`` - labels the elements by their expression 

as a product of cycles 

 

EXAMPLES:: 

 

sage: posets.SymmetricGroupAbsoluteOrderPoset(4) 

Finite poset containing 24 elements 

sage: posets.SymmetricGroupAbsoluteOrderPoset(3, labels="cycles") 

Finite poset containing 6 elements 

sage: posets.SymmetricGroupAbsoluteOrderPoset(3, labels="reduced_words") 

Finite poset containing 6 elements 

""" 

from sage.groups.perm_gps.permgroup_named import SymmetricGroup 

W = SymmetricGroup(n) 

if labels == "permutations": 

element_labels = {s: s.tuple() for s in W} 

if labels == "reduced_words": 

element_labels = {s: tuple(s.reduced_word()) for s in W} 

if labels == "cycles": 

element_labels = {s: "".join(x for x in s.cycle_string() if x != ',') 

for s in W} 

 

return Poset({s: s.absolute_covers() for s in W}, element_labels) 

 

@staticmethod 

def UpDownPoset(n, m=1): 

r""" 

Return the up-down poset on `n` elements where every `(m+1)` 

step is down and the rest are up. 

 

The case where `m=1` is sometimes referred to as the zig-zag poset 

or the fence. 

 

INPUT: 

 

- ``n`` - nonnegative integer, number of elements in the poset 

- ``m`` - nonnegative integer (default 1), how frequently down 

steps occur 

 

OUTPUT: 

 

The partially ordered set on `\{ 0, 1, \ldots, n-1 \}` 

where `i` covers `i+1` if `m` divides `i+1`, and `i+1` covers `i` 

otherwise. 

 

EXAMPLES:: 

 

sage: P = posets.UpDownPoset(7, 2); P 

Finite poset containing 7 elements 

sage: sorted(P.cover_relations()) 

[[0, 1], [1, 2], [3, 2], [3, 4], [4, 5], [6, 5]] 

 

Fibonacci numbers as the number of antichains of a poset:: 

 

sage: [len(posets.UpDownPoset(n).antichains().list()) for n in range(6)] 

[1, 2, 3, 5, 8, 13] 

 

TESTS:: 

 

sage: P = posets.UpDownPoset(0); P 

Finite poset containing 0 elements 

""" 

try: 

n = Integer(n) 

except TypeError: 

raise TypeError("number of elements must be an integer, not {0}".format(n)) 

if n < 0: 

raise ValueError("number of elements must be non-negative, not {0}".format(n)) 

try: 

m = Integer(m) 

except TypeError: 

raise TypeError("parameter m must be an integer, not {0}".format(m)) 

if m < 1: 

raise ValueError("parameter m must be positive, not {0}".format(m)) 

 

covers = [[i, i + 1] if (i + 1) % (m + 1) else [i + 1, i] 

for i in range(n - 1)] 

return Poset((range(n), covers), cover_relations=True) 

 

@staticmethod 

def YoungDiagramPoset(lam): 

""" 

Return the poset of cells in the Young diagram of a partition. 

 

INPUT: 

 

- ``lam`` -- a partition 

 

EXAMPLES:: 

 

sage: P = posets.YoungDiagramPoset(Partition([2,2])); P 

Finite meet-semilattice containing 4 elements 

sage: P.cover_relations() 

[[(0, 0), (0, 1)], [(0, 0), (1, 0)], [(0, 1), (1, 1)], [(1, 0), 

(1, 1)]] 

""" 

def cell_leq(a, b): 

""" 

Nested function that returns `True` if the cell `a` is 

to the left or above 

the cell `b` in the (English) Young diagram. 

""" 

return ((a[0] == b[0] - 1 and a[1] == b[1]) or 

(a[1] == b[1] - 1 and a[0] == b[0])) 

return MeetSemilattice((lam.cells(), cell_leq), cover_relations=True) 

 

@staticmethod 

def YoungsLattice(n): 

""" 

Return Young's Lattice up to rank `n`. 

 

In other words, the poset of partitions 

of size less than or equal to `n` ordered by inclusion. 

 

INPUT: 

 

- ``n`` -- a positive integer 

 

EXAMPLES:: 

 

sage: P = posets.YoungsLattice(3); P 

Finite meet-semilattice containing 7 elements 

sage: P.cover_relations() 

[[[], [1]], 

[[1], [1, 1]], 

[[1], [2]], 

[[1, 1], [1, 1, 1]], 

[[1, 1], [2, 1]], 

[[2], [2, 1]], 

[[2], [3]]] 

""" 

from sage.combinat.partition import Partitions, Partition 

from sage.misc.flatten import flatten 

partitions = flatten([list(Partitions(i)) for i in range(n + 1)]) 

return JoinSemilattice((partitions, Partition.contains)).dual() 

 

@staticmethod 

def YoungsLatticePrincipalOrderIdeal(lam): 

""" 

Return the principal order ideal of the 

partition `lam` in Young's Lattice. 

 

INPUT: 

 

- ``lam`` -- a partition 

 

EXAMPLES:: 

 

sage: P = posets.YoungsLatticePrincipalOrderIdeal(Partition([2,2])) 

sage: P 

Finite lattice containing 6 elements 

sage: P.cover_relations() 

[[[], [1]], 

[[1], [1, 1]], 

[[1], [2]], 

[[1, 1], [2, 1]], 

[[2], [2, 1]], 

[[2, 1], [2, 2]]] 

""" 

from sage.misc.flatten import flatten 

from sage.combinat.partition import Partition 

 

def lower_covers(l): 

""" 

Nested function returning those partitions obtained 

from the partition `l` by removing 

a single cell. 

""" 

return [l.remove_cell(c[0], c[1]) for c in l.removable_cells()] 

 

def contained_partitions(l): 

""" 

Nested function returning those partitions contained in 

the partition `l` 

""" 

if l == Partition([]): 

return l 

return flatten([l, [contained_partitions(m) 

for m in lower_covers(l)]]) 

 

ideal = list(set(contained_partitions(lam))) 

H = DiGraph(dict([[p, lower_covers(p)] for p in ideal])) 

return LatticePoset(H.reverse()) 

 

## RANDOM LATTICES 

 

# Following are helper functions for random lattice generation. 

# There is no parameter checking, 0, 1, ..., n may or may not be a 

# linear extension, exact output type may vary, etc. Direct use is 

# discouraged. Use by posets.RandomLattice(..., properties=[...]). 

 

 

def _random_lattice(n, p): 

""" 

Return a random lattice. 

 

INPUT: 

 

- ``n`` -- number of elements, a non-negative integer 

- ``p`` -- a number at least zero and less than one; higher number 

means more covering relations 

 

OUTPUT: 

 

A list of lists. Interpreted as a list of lower covers 

for a poset, it is a lattice with ``0..n-1`` as a linear 

extension. 

 

EXAMPLES:: 

 

sage: set_random_seed(42) # Results are reproducible 

sage: sage.combinat.posets.poset_examples._random_lattice(7, 0.4) 

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

 

ALGORITHM:: 

 

We add elements one by one. We check that adding a maximal 

element `e` to a meet-semilattice `L` with maximal elements 

`M` will create a semilattice by checking that there is a 

meet for `e, m` for all `m \in M`. We do that by keeping 

track of meet matrix and list of maximal elements. 

""" 

from sage.functions.other import floor, sqrt 

from sage.misc.prandom import random 

 

n = n-1 

meets = [[None]*n for _ in range(n)] 

meets[0][0] = 0 

maxs = set([0]) 

lc_all = [[]] # No lower covers for the bottom element. 

 

for i in range(1, n): 

 

# First add some random element as a lower cover. 

# Alone it can't change a semilattice to non-semilattice, 

# so we don't check it. 

new = i-1-floor(i*sqrt(random())) 

lc_list = [new] 

maxs.discard(new) 

max_meets = {m:meets[m][new] for m in maxs} 

 

while random() < p and 0 not in lc_list: 

# An ad hoc solution. srqt(random()) instead of randint(0, i) 

# make number of coatoms closer to number of atoms. 

new = i-1-floor(i*sqrt(random())) 

 

# Check that lc_list + new is an antichain. 

if any(meets[new][lc] in [new, lc] for lc in lc_list): 

continue 

 

# Check that new has a unique meet with any maximal element. 

for m in maxs: 

meet_m = meets[m][new] 

if meets[meet_m][max_meets[m]] not in [meet_m, max_meets[m]]: 

break 

 

else: # So, we found a new lower cover for i. 

lc_list.append(new) 

for m in maxs: 

max_meets[m] = max(max_meets[m], meets[m][new]) 

maxs.discard(new) 

 

# Now compute new row and column to meet matrix. 

meets[i][i] = i 

for lc in lc_list: 

meets[i][lc] = meets[lc][i] = lc 

for e in range(i): 

meets[i][e] = meets[e][i] = max(meets[e][lc] for lc in lc_list) 

 

maxs.add(i) 

lc_all.append(lc_list) 

 

lc_all.append(list(maxs)) # Add the top element. 

return lc_all 

 

 

def _random_dismantlable_lattice(n): 

""" 

Return a random dismantlable lattice on `n` elements. 

 

INPUT: 

 

- ``n`` -- number of elements, a non-negative integer 

 

OUTPUT: 

 

A digraph that can be interpreted as the Hasse diagram of a random 

dismantlable lattice. It has `0` as the bottom element and `n-1` as 

the top element, but otherwise `0, \ldots, n-1` *is not* usually a 

linear extension of the lattice. 

 

EXAMPLES:: 

 

sage: set_random_seed(78) # Results are reproducible 

sage: D = sage.combinat.posets.poset_examples._random_dismantlable_lattice(10); D 

Digraph on 10 vertices 

sage: D.neighbors_in(8) 

[0] 

 

ALGORITHM:: 

 

We add elements one by one by "de-dismantling", i.e. select 

a random pair of comparable elements and add a new element 

between them. 

""" 

from sage.misc.prandom import randint 

 

D = DiGraph({0: [n-1]}) 

for i in range(1, n-1): 

a = randint(0, i//2) 

b_ = list(D.depth_first_search(a)) 

b = b_[randint(1, len(b_)-1)] 

D.add_vertex(i) 

D.add_edge(a, i) 

D.add_edge(i, b) 

D.delete_edge(a, b) 

return D 

 

 

def _random_planar_lattice(n): 

""" 

Return a random planar lattice on `n` elements. 

 

INPUT: 

 

- ``n`` -- number of elements, a non-negative integer 

 

OUTPUT: 

 

A random planar lattice. It has `0` as the bottom 

element and `n-1` as the top element, but otherwise 

`0, \ldots, n-1` *is not* usually a linear extension of 

the lattice. 

 

EXAMPLES:: 

 

sage: set_random_seed(78) # Results are reproducible 

sage: D = sage.combinat.posets.poset_examples._random_planar_lattice(10); D 

Digraph on 10 vertices 

sage: D.neighbors_in(8) 

[1] 

 

ALGORITHM:: 

 

Every planar lattice is dismantlable. 

 

We add elements one by one like when generating 

dismantlable lattices, and after every addition 

check that we still have a planar lattice. 

""" 

from sage.misc.prandom import randint 

 

G = DiGraph({0: [n-1]}) 

while G.order() < n: 

i = G.order()-1 

a = randint(0, i//2) 

b_ = list(G.depth_first_search(a)) 

b = b_[randint(1, len(b_)-1)] 

G1 = G.copy() 

G.add_vertex(i) 

G.add_edge(a, i) 

G.add_edge(i, b) 

G.delete_edge(a, b) 

G2 = G.copy() 

G2.add_edge(n-1, 0) 

if not G2.is_planar(): 

G = G1.copy() 

return G 

 

 

def _random_distributive_lattice(n): 

""" 

Return a random poset that has `n` antichains. 

 

INPUT: 

 

- ``n`` -- number of elements, a non-negative integer 

 

OUTPUT: 

 

A random poset (as DiGraph) that has `n` antichains; i.e. a poset 

that's order ideals lattice has `n` elements. 

 

EXAMPLES:: 

 

sage: g = sage.combinat.posets.poset_examples._random_distributive_lattice(10) 

sage: Poset(g).order_ideals_lattice(as_ideals=False).cardinality() 

10 

 

ALGORITHM: 

 

Add elements until there are at least `n` antichains. 

Remove elements until there are at most `n` antichains. 

Repeat. 

""" 

from sage.combinat.posets.hasse_diagram import HasseDiagram 

from copy import copy 

from sage.combinat.subset import Subsets 

from sage.graphs.digraph_generators import digraphs 

 

if n < 4: 

return digraphs.Path(n-1) 

 

H = HasseDiagram({0: []}) 

while sum(1 for _ in H.antichains_iterator()) < n: 

D = copy(H) 

newcover = Subsets(H).random_element() 

new_element = H.order() 

D.add_vertex(new_element) 

for e in newcover: 

D.add_edge(e, new_element) 

 

D = D.transitive_reduction() 

H = HasseDiagram(D) 

 

while sum(1 for _ in H.antichains_iterator()) > n: 

D = copy(H) 

to_delete = H.random_vertex() 

for a in D.neighbors_in(to_delete): 

for b in D.neighbors_out(to_delete): 

D.add_edge(a, b) 

D.delete_vertex(to_delete) 

D.relabel({z:z-1 for z in range(to_delete + 1, D.order() + 1)}) 

H = HasseDiagram(D) 

return D 

 

def _random_stone_lattice(n): 

""" 

Return a random Stone lattice on `n` elements. 

 

INPUT: 

 

- ``n`` -- number of elements, a non-negative integer 

 

OUTPUT: 

 

A random lattice (as a digraph) of `n` elements. 

 

EXAMPLES:: 

 

sage: g = sage.combinat.posets.poset_examples._random_stone_lattice(10) 

sage: LatticePoset(g).is_stone() 

True 

 

ALGORITHM: 

 

Randomly split `n` to some factors. For every factor `p` generate 

a random distributive lattice on `p-1` elements and add a new bottom 

element to it. Compute the cartesian product of those lattices. 

""" 

from sage.arith.misc import factor 

from sage.combinat.partition import Partitions 

from sage.misc.misc_c import prod 

from copy import copy 

 

factors = sum([[f[0]]*f[1] for f in factor(n)], []) 

sage.misc.prandom.shuffle(factors) 

 

part_lengths = list(Partitions(len(factors)).random_element()) 

parts = [] 

while part_lengths: 

x = part_lengths.pop() 

parts.append(prod(factors[:x])) 

factors = factors[x:] 

 

result = DiGraph(1) 

for p in parts: 

g = _random_distributive_lattice(p-1) 

g = copy(Poset(g).order_ideals_lattice(as_ideals=False)._hasse_diagram) 

g.add_edge('bottom', 0) 

result = result.cartesian_product(g) 

result.relabel() 

 

return result 

 

posets = Posets