Returns the species of permutations.
EXAMPLES:
sage: P = species.PermutationSpecies()
sage: P.generating_series().coefficients(5)
[1, 1, 1, 1, 1]
sage: P.isotype_generating_series().coefficients(5)
[1, 1, 2, 3, 5]
Bases: sage.combinat.species.structure.GenericSpeciesStructure
Returns the group of permutations whose action on this structure leave it fixed.
EXAMPLES:
sage: p = PermutationGroupElement((2,3,4))
sage: P = species.PermutationSpecies()
sage: a = P.structures(["a", "b", "c", "d"]).random_element(); a
['a', 'c', 'b', 'd']
sage: a.automorphism_group()
Permutation Group with generators [(2,3), (1,4)]
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']]
EXAMPLES:
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']]
Returns self as a permutation group element.
EXAMPLES:
sage: p = PermutationGroupElement((2,3,4))
sage: P = species.PermutationSpecies()
sage: a = P.structures(["a", "b", "c", "d"]).random_element(); a
['a', 'c', 'b', 'd']
sage: a.permutation_group_element()
(2,3)
Returns the transport of this structure along the permutation perm.
EXAMPLES:
sage: p = PermutationGroupElement((2,3,4))
sage: P = species.PermutationSpecies()
sage: a = P.structures(["a", "b", "c", "d"]).random_element(); a
['a', 'c', 'b', 'd']
sage: a.transport(p)
['a', 'd', 'c', 'b']