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# Partition Species

class `sage.combinat.species.partition_species.``PartitionSpecies`(min=None, max=None, weight=None)

  Bases: `sage.combinat.species.species.GenericCombinatorialSpecies`

  Returns the species of partitions.

  EXAMPLES:

      sage: P = species.PartitionSpecies()
      sage: P.generating_series().coefficients(5)
      [1, 1, 1, 5/6, 5/8]
      sage: P.isotype_generating_series().coefficients(5)
      [1, 1, 2, 3, 5]

      sage: P = species.PartitionSpecies()
      sage: P._check()
      True
      sage: P == loads(dumps(P))
      True

class `sage.combinat.species.partition_species.``PartitionSpeciesStructure`(parent, labels, list)

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

  EXAMPLES:

      sage: from sage.combinat.species.partition_species import PartitionSpeciesStructure
      sage: P = species.PartitionSpecies()
      sage: s = PartitionSpeciesStructure(P, ['a','b','c'], [[1,2],[3]]); s
      {{'a', 'b'}, {'c'}}
      sage: s == loads(dumps(s))
      True

  `automorphism_group`()

    Returns the group of permutations whose action on this set partition leave it fixed.

    EXAMPLES:

        sage: p = PermutationGroupElement((2,3))
        sage: from sage.combinat.species.partition_species import PartitionSpeciesStructure
        sage: a = PartitionSpeciesStructure(None, [2,3,4], [[1,2],[3]]); a
        {{2, 3}, {4}}
        sage: a.automorphism_group()
        Permutation Group with generators [(1,2)]

  `canonical_label`()

    EXAMPLES:

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

  `change_labels`(labels)

    EXAMPLES:

        sage: p = PermutationGroupElement((2,3))
        sage: from sage.combinat.species.partition_species import PartitionSpeciesStructure
        sage: a = PartitionSpeciesStructure(None, [2,3,4], [[1,2],[3]]); a
        {{2, 3}, {4}}
        sage: a.change_labels([1,2,3])
        {{1, 2}, {3}}

  `transport`(perm)

    Returns the transport of this set partition along the permutation perm. For set partitions, this is the direct product of the automorphism groups for each of the blocks.

    EXAMPLES:

        sage: p = PermutationGroupElement((2,3))
        sage: from sage.combinat.species.partition_species import PartitionSpeciesStructure
        sage: a = PartitionSpeciesStructure(None, [2,3,4], [[1,2],[3]]); a
        {{2, 3}, {4}}
        sage: a.transport(p)
        {{2, 4}, {3}}

`sage.combinat.species.partition_species.``PartitionSpecies_class`

  alias of `PartitionSpecies`

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