Partition Species

sage.combinat.species.partition_species.PartitionSpecies(*args, **kwds)

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]

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

Bases: sage.combinat.species.structure.GenericSpeciesStructure

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}}

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

Bases: sage.combinat.species.species.GenericCombinatorialSpecies