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# Characteristic Species

class `sage.combinat.species.characteristic_species.``CharacteristicSpecies`(n, min=None, max=None, weight=None)

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

  Returns the characteristic species of order \(n\).

  This species has exactly one structure on a set of of size \(n\) and no structures of on sets of any other size.

  EXAMPLES:

      sage: X = species.CharacteristicSpecies(1)
      sage: X.structures([1]).list()
      [1]
      sage: X.structures([1,2]).list()
      []
      sage: X.generating_series().coefficients(4)
      [0, 1, 0, 0]
      sage: X.isotype_generating_series().coefficients(4)
      [0, 1, 0, 0]
      sage: X.cycle_index_series().coefficients(4)
      [0, p[1], 0, 0]

      sage: F = species.CharacteristicSpecies(3)
      sage: c = F.generating_series().coefficients(4)
      sage: F._check()
      True
      sage: F == loads(dumps(F))
      True

  TESTS:

      sage: S1 = species.CharacteristicSpecies(1)
      sage: S2 = species.CharacteristicSpecies(1)
      sage: S3 = species.CharacteristicSpecies(2)
      sage: S4 = species.CharacteristicSpecies(2, weight=2)
      sage: S1 is S2
      True
      sage: S1 == S3
      False

class `sage.combinat.species.characteristic_species.``CharacteristicSpeciesStructure`(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. For the characteristic species, there is only one structure, so every permutation is in its automorphism group.

    EXAMPLES:

        sage: F = species.CharacteristicSpecies(3)
        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

  `canonical_label`()

    EXAMPLES:

        sage: F = species.CharacteristicSpecies(3)
        sage: a = F.structures(["a", "b", "c"]).random_element(); a
        {'a', 'b', 'c'}
        sage: a.canonical_label()
        {'a', 'b', 'c'}

  `transport`(perm)

    Returns the transport of this structure along the permutation perm.

    EXAMPLES:

        sage: F = species.CharacteristicSpecies(3)
        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'}

`sage.combinat.species.characteristic_species.``CharacteristicSpecies_class`

  alias of `CharacteristicSpecies`

class `sage.combinat.species.characteristic_species.``EmptySetSpecies`(min=None, max=None, weight=None)

  Bases: `sage.combinat.species.characteristic_species.CharacteristicSpecies`

  Returns the empty set species.

  This species has exactly one structure on the empty set. It is the same (and is implemented) as `CharacteristicSpecies(0)`.

  EXAMPLES:

      sage: X = species.EmptySetSpecies()
      sage: X.structures([]).list()
      [{}]
      sage: X.structures([1,2]).list()
      []
      sage: X.generating_series().coefficients(4)
      [1, 0, 0, 0]
      sage: X.isotype_generating_series().coefficients(4)
      [1, 0, 0, 0]
      sage: X.cycle_index_series().coefficients(4)
      [p[], 0, 0, 0]

  TESTS:

      sage: E1 = species.EmptySetSpecies()
      sage: E2 = species.EmptySetSpecies()
      sage: E1 is E2
      True

      sage: E = species.EmptySetSpecies()
      sage: E._check()
      True
      sage: E == loads(dumps(E))
      True

`sage.combinat.species.characteristic_species.``EmptySetSpecies_class`

  alias of `EmptySetSpecies`

class `sage.combinat.species.characteristic_species.``SingletonSpecies`(min=None, max=None, weight=None)

  Bases: `sage.combinat.species.characteristic_species.CharacteristicSpecies`

  Returns the species of singletons.

  This species has exactly one structure on a set of size \(1\). It is the same (and is implemented) as `CharacteristicSpecies(1)`.

  EXAMPLES:

      sage: X = species.SingletonSpecies()
      sage: X.structures([1]).list()
      [1]
      sage: X.structures([1,2]).list()
      []
      sage: X.generating_series().coefficients(4)
      [0, 1, 0, 0]
      sage: X.isotype_generating_series().coefficients(4)
      [0, 1, 0, 0]
      sage: X.cycle_index_series().coefficients(4)
      [0, p[1], 0, 0]

  TESTS:

      sage: S1 = species.SingletonSpecies()
      sage: S2 = species.SingletonSpecies()
      sage: S1 is S2
      True

      sage: S = species.SingletonSpecies()
      sage: S._check()
      True
      sage: S == loads(dumps(S))
      True

`sage.combinat.species.characteristic_species.``SingletonSpecies_class`

  alias of `SingletonSpecies`

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