Permutation species

class sage.combinat.species.permutation_species.PermutationSpecies(min=None, max=None, weight=None)

Bases: GenericCombinatorialSpecies, UniqueRepresentation

Returns the species of permutations.

EXAMPLES:

Sage

sage: P = species.PermutationSpecies()
sage: P.generating_series()[0:5]
[1, 1, 1, 1, 1]
sage: P.isotype_generating_series()[0:5]
[1, 1, 2, 3, 5]

sage: P = species.PermutationSpecies()
sage: c = P.generating_series()[0:3]
sage: P._check()
True
sage: P == loads(dumps(P))
True

Python

>>> from sage.all import *
>>> P = species.PermutationSpecies()
>>> P.generating_series()[Integer(0):Integer(5)]
[1, 1, 1, 1, 1]
>>> P.isotype_generating_series()[Integer(0):Integer(5)]
[1, 1, 2, 3, 5]

>>> P = species.PermutationSpecies()
>>> c = P.generating_series()[Integer(0):Integer(3)]
>>> P._check()
True
>>> P == loads(dumps(P))
True

class sage.combinat.species.permutation_species.PermutationSpeciesStructure(parent, labels, list)

Bases: GenericSpeciesStructure

automorphism_group()

Returns the group of permutations whose action on this structure leave it fixed.

EXAMPLES:

Sage

sage: set_random_seed(0)
sage: p = PermutationGroupElement((2,3,4))
sage: P = species.PermutationSpecies()
sage: a = P.structures(["a", "b", "c", "d"])[2]; a
['a', 'c', 'b', 'd']
sage: a.automorphism_group()
Permutation Group with generators [(2,3), (1,4)]

Python

>>> from sage.all import *
>>> set_random_seed(Integer(0))
>>> p = PermutationGroupElement((Integer(2),Integer(3),Integer(4)))
>>> P = species.PermutationSpecies()
>>> a = P.structures(["a", "b", "c", "d"])[Integer(2)]; a
['a', 'c', 'b', 'd']
>>> a.automorphism_group()
Permutation Group with generators [(2,3), (1,4)]

Sage

sage: [a.transport(perm) for perm in a.automorphism_group()]
[['a', 'c', 'b', 'd'],
 ['a', 'c', 'b', 'd'],
 ['a', 'c', 'b', 'd'],
 ['a', 'c', 'b', 'd']]

Python

>>> from sage.all import *
>>> [a.transport(perm) for perm in a.automorphism_group()]
[['a', 'c', 'b', 'd'],
 ['a', 'c', 'b', 'd'],
 ['a', 'c', 'b', 'd'],
 ['a', 'c', 'b', 'd']]

canonical_label()

EXAMPLES:

Sage

sage: P = species.PermutationSpecies()
sage: S = P.structures(["a", "b", "c"])
sage: [s.canonical_label() for s in S]
[['a', 'b', 'c'],
 ['b', 'a', 'c'],
 ['b', 'a', 'c'],
 ['b', 'c', 'a'],
 ['b', 'c', 'a'],
 ['b', 'a', 'c']]

Python

>>> from sage.all import *
>>> P = species.PermutationSpecies()
>>> S = P.structures(["a", "b", "c"])
>>> [s.canonical_label() for s in S]
[['a', 'b', 'c'],
 ['b', 'a', 'c'],
 ['b', 'a', 'c'],
 ['b', 'c', 'a'],
 ['b', 'c', 'a'],
 ['b', 'a', 'c']]

permutation_group_element()

Returns self as a permutation group element.

EXAMPLES:

Sage

sage: p = PermutationGroupElement((2,3,4))
sage: P = species.PermutationSpecies()
sage: a = P.structures(["a", "b", "c", "d"])[2]; a
['a', 'c', 'b', 'd']
sage: a.permutation_group_element()
(2,3)

Python

>>> from sage.all import *
>>> p = PermutationGroupElement((Integer(2),Integer(3),Integer(4)))
>>> P = species.PermutationSpecies()
>>> a = P.structures(["a", "b", "c", "d"])[Integer(2)]; a
['a', 'c', 'b', 'd']
>>> a.permutation_group_element()
(2,3)

transport(perm)

Returns the transport of this structure along the permutation perm.

EXAMPLES:

Sage

sage: p = PermutationGroupElement((2,3,4))
sage: P = species.PermutationSpecies()
sage: a = P.structures(["a", "b", "c", "d"])[2]; a
['a', 'c', 'b', 'd']
sage: a.transport(p)
['a', 'd', 'c', 'b']

Python

>>> from sage.all import *
>>> p = PermutationGroupElement((Integer(2),Integer(3),Integer(4)))
>>> P = species.PermutationSpecies()
>>> a = P.structures(["a", "b", "c", "d"])[Integer(2)]; a
['a', 'c', 'b', 'd']
>>> a.transport(p)
['a', 'd', 'c', 'b']

sage.combinat.species.permutation_species.PermutationSpecies_class

alias of PermutationSpecies