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# Permutation species

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

  Bases: `sage.combinat.species.species.GenericCombinatorialSpecies`, `sage.structure.unique_representation.UniqueRepresentation`

  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]

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

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

  Bases: `sage.combinat.species.structure.GenericSpeciesStructure`

  EXAMPLES:

      sage: from sage.combinat.species.structure import GenericSpeciesStructure
      sage: a = GenericSpeciesStructure(None, [2,3,4], [1,2,3])
      sage: a
      [2, 3, 4]
      sage: a.parent() is None
      True
      sage: a == loads(dumps(a))
      True

  `automorphism_group`()

    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']]

  `canonical_label`()

    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']]

  `permutation_group_element`()

    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)

  `transport`(perm)

    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']

`sage.combinat.species.permutation_species.``PermutationSpecies_class`

  alias of `PermutationSpecies`

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