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# Subset Species

class `sage.combinat.species.subset_species.``SubsetSpecies`(min=None, max=None, weight=None)

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

  Returns the species of subsets.

  EXAMPLES:

      sage: S = species.SubsetSpecies()
      sage: S.generating_series().coefficients(5)
      [1, 2, 2, 4/3, 2/3]
      sage: S.isotype_generating_series().coefficients(5)
      [1, 2, 3, 4, 5]

      sage: S = species.SubsetSpecies()
      sage: c = S.generating_series().coefficients(3)
      sage: S._check()
      True
      sage: S == loads(dumps(S))
      True

class `sage.combinat.species.subset_species.``SubsetSpeciesStructure`(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 subset leave it fixed.

    EXAMPLES:

        sage: F = species.SubsetSpecies()
        sage: a = F.structures([1,2,3,4])[6]; a
        {1, 3}
        sage: a.automorphism_group()
        Permutation Group with generators [(2,4), (1,3)]

        sage: [a.transport(g) for g in a.automorphism_group()]
        [{1, 3}, {1, 3}, {1, 3}, {1, 3}]

  `canonical_label`()

    EXAMPLES:

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

  `complement`()

    EXAMPLES:

        sage: F = species.SubsetSpecies()
        sage: a = F.structures(["a", "b", "c"])[5]; a
        {'a', 'c'}
        sage: a.complement()
        {'b'}

  `labels`()

    EXAMPLES:

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

  `transport`(perm)

    Returns the transport of this subset along the permutation perm.

    EXAMPLES:

        sage: F = species.SubsetSpecies()
        sage: a = F.structures(["a", "b", "c"])[5]; a
        {'a', 'c'}
        sage: p = PermutationGroupElement((1,2))
        sage: a.transport(p)
        {'b', 'c'}
        sage: p = PermutationGroupElement((1,3))
        sage: a.transport(p)
        {'a', 'c'}

`sage.combinat.species.subset_species.``SubsetSpecies_class`

  alias of `SubsetSpecies`

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