Coverage for local/lib/python2.7/site-packages/sage/combinat/species/set_species.py : 47%

""" 

Set Species 

""" 

from __future__ import absolute_import 

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

# Copyright (C) 2008 Mike Hansen <mhansen@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 six.moves import range 

from .species import GenericCombinatorialSpecies 

from .generating_series import factorial_stream, _integers_from 

from sage.combinat.species.structure import GenericSpeciesStructure 

from sage.combinat.species.misc import accept_size 

from sage.structure.unique_representation import UniqueRepresentation 

class SetSpeciesStructure(GenericSpeciesStructure): 

def __repr__(self): 

""" 

EXAMPLES:: 

sage: S = species.SetSpecies() 

sage: a = S.structures(["a","b","c"]).random_element(); a 

{'a', 'b', 'c'} 

""" 

s = GenericSpeciesStructure.__repr__(self) 

return "{"+s[1:-1]+"}" 

def canonical_label(self): 

""" 

EXAMPLES:: 

sage: S = species.SetSpecies() 

sage: a = S.structures(["a","b","c"]).random_element(); a 

{'a', 'b', 'c'} 

sage: a.canonical_label() 

{'a', 'b', 'c'} 

""" 

rng = list(range(1, len(self._labels) + 1)) 

return SetSpeciesStructure(self.parent(), self._labels, rng) 

def transport(self, perm): 

""" 

Returns the transport of this set along the permutation perm. 

EXAMPLES:: 

sage: F = species.SetSpecies() 

sage: a = F.structures(["a", "b", "c"]).random_element(); a 

{'a', 'b', 'c'} 

sage: p = PermutationGroupElement((1,2)) 

sage: a.transport(p) 

{'a', 'b', 'c'} 

""" 

return self 

def automorphism_group(self): 

""" 

Returns the group of permutations whose action on this set leave it 

fixed. For the species of sets, there is only one isomorphism 

class, so every permutation is in its automorphism group. 

EXAMPLES:: 

sage: F = species.SetSpecies() 

sage: a = F.structures(["a", "b", "c"]).random_element(); a 

{'a', 'b', 'c'} 

sage: a.automorphism_group() 

Symmetric group of order 3! as a permutation group 

""" 

from sage.groups.all import SymmetricGroup 

return SymmetricGroup(max(1,len(self._labels))) 

class SetSpecies(GenericCombinatorialSpecies, UniqueRepresentation): 

@staticmethod 

@accept_size 

def __classcall__(cls, *args, **kwds): 

""" 

EXAMPLES:: 

sage: E = species.SetSpecies(); E 

Set species 

""" 

return super(SetSpecies, cls).__classcall__(cls, *args, **kwds) 

def __init__(self, min=None, max=None, weight=None): 

""" 

Returns the species of sets. 

EXAMPLES:: 

sage: E = species.SetSpecies() 

sage: E.structures([1,2,3]).list() 

[{1, 2, 3}] 

sage: E.isotype_generating_series().coefficients(4) 

[1, 1, 1, 1] 

sage: S = species.SetSpecies() 

sage: c = S.generating_series().coefficients(3) 

sage: S._check() 

True 

sage: S == loads(dumps(S)) 

True 

""" 

GenericCombinatorialSpecies.__init__(self, min=min, max=max, weight=weight) 

self._name = "Set species" 

_default_structure_class = SetSpeciesStructure 

def _structures(self, structure_class, labels): 

""" 

EXAMPLES:: 

sage: S = species.SetSpecies() 

sage: S.structures([1,2,3]).list() 

[{1, 2, 3}] 

""" 

n = len(labels) 

yield structure_class(self, labels, range(1,n+1)) 

_isotypes = _structures 

def _gs_iterator(self, base_ring): 

r""" 

The generating series for the species of sets is given by 

`e^x`. 

EXAMPLES:: 

sage: S = species.SetSpecies() 

sage: g = S.generating_series() 

sage: g.coefficients(10) 

[1, 1, 1/2, 1/6, 1/24, 1/120, 1/720, 1/5040, 1/40320, 1/362880] 

sage: [g.count(i) for i in range(10)] 

[1, 1, 1, 1, 1, 1, 1, 1, 1, 1] 

""" 

for n in _integers_from(0): 

yield base_ring(self._weight/factorial_stream[n]) 

def _itgs_list(self, base_ring): 

r""" 

The isomorphism type generating series for the species of sets is 

`\frac{1}{1-x}`. 

EXAMPLES:: 

sage: S = species.SetSpecies() 

sage: g = S.isotype_generating_series() 

sage: g.coefficients(10) 

[1, 1, 1, 1, 1, 1, 1, 1, 1, 1] 

sage: [g.count(i) for i in range(10)] 

[1, 1, 1, 1, 1, 1, 1, 1, 1, 1] 

""" 

return [base_ring(self._weight)] 

def _cis(self, series_ring, base_ring): 

r""" 

The cycle index series for the species of sets is given by 

`exp\( \sum_{n=1}{\infty} = \frac{x_n}{n} \)`. 

EXAMPLES:: 

sage: S = species.SetSpecies() 

sage: g = S.cycle_index_series() 

sage: g.coefficients(5) 

[p[], 

p[1], 

1/2*p[1, 1] + 1/2*p[2], 

1/6*p[1, 1, 1] + 1/2*p[2, 1] + 1/3*p[3], 

1/24*p[1, 1, 1, 1] + 1/4*p[2, 1, 1] + 1/8*p[2, 2] + 1/3*p[3, 1] + 1/4*p[4]] 

""" 

from .generating_series import ExponentialCycleIndexSeries 

res = ExponentialCycleIndexSeries(base_ring) 

if self.is_weighted(): 

res *= self._weight 

return res 

#Backward compatibility 

SetSpecies_class = SetSpecies