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# Cycle Species

class `sage.combinat.species.cycle_species.``CycleSpecies`(min=None, max=None, weight=None)

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

  Returns the species of cycles.

  EXAMPLES:

      sage: C = species.CycleSpecies(); C
      Cyclic permutation species
      sage: C.structures([1,2,3,4]).list()
      [(1, 2, 3, 4),
       (1, 2, 4, 3),
       (1, 3, 2, 4),
       (1, 3, 4, 2),
       (1, 4, 2, 3),
       (1, 4, 3, 2)]

  TESTS: We check to verify that the caching of species is actually working.

      sage: species.CycleSpecies() is species.CycleSpecies()
      True

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

class `sage.combinat.species.cycle_species.``CycleSpeciesStructure`(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 = species.CycleSpecies()
        sage: a = P.structures([1, 2, 3, 4]).random_element(); a
        (1, 2, 3, 4)
        sage: a.automorphism_group()
        Permutation Group with generators [(1,2,3,4)]

        sage: [a.transport(perm) for perm in a.automorphism_group()]
        [(1, 2, 3, 4), (1, 2, 3, 4), (1, 2, 3, 4), (1, 2, 3, 4)]

  `canonical_label`()

    EXAMPLES:

        sage: P = species.CycleSpecies()
        sage: P.structures(["a","b","c"]).random_element().canonical_label()
        ('a', 'b', 'c')

  `permutation_group_element`()

    Returns this cycle as a permutation group element.

    EXAMPLES:

        sage: F = species.CycleSpecies()
        sage: a = F.structures(["a", "b", "c"]).random_element(); a
        ('a', 'b', 'c')
        sage: a.permutation_group_element()
        (1,2,3)

  `transport`(perm)

    Returns the transport of this structure along the permutation perm.

    EXAMPLES:

        sage: F = species.CycleSpecies()
        sage: a = F.structures(["a", "b", "c"]).random_element(); a
        ('a', 'b', 'c')
        sage: p = PermutationGroupElement((1,2))
        sage: a.transport(p)
        ('a', 'c', 'b')

`sage.combinat.species.cycle_species.``CycleSpecies_class`

  alias of `CycleSpecies`

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