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]
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
EXAMPLES:
sage: S = species.PartitionSpecies()
sage: a = S.structures(["a","b","c"]).random_element(); a
{{'a', 'b', 'c'}}
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)]
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'}}]
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}}
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}}
EXAMPLES:
sage: P = species.PartitionSpecies()
sage: P._check()
True
sage: P == loads(dumps(P))
True
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]
Returns the canonical representative corresponding to the partition p.
EXAMPLES:
sage: P = species.PartitionSpecies()
sage: P._canonical_rep_from_partition(P._default_structure_class,[1,2,3],[2,1])
{{1, 2}, {3}}
The cycle index series for the species of partitions is given by
EXAMPLES:
sage: P = species.PartitionSpecies()
sage: g = P.cycle_index_series()
sage: g.coefficients(5)
[p[],
p[1],
p[1, 1] + p[2],
5/6*p[1, 1, 1] + 3/2*p[2, 1] + 2/3*p[3],
5/8*p[1, 1, 1, 1] + 7/4*p[2, 1, 1] + 7/8*p[2, 2] + p[3, 1] + 3/4*p[4]]
EXAMPLES:
sage: P = species.PartitionSpecies()
sage: g = P.generating_series()
sage: g.coefficients(5)
[1, 1, 1, 5/6, 5/8]
EXAMPLES:
sage: P = species.PartitionSpecies()
sage: P.isotypes([1,2,3,4]).list()
[{{1, 2, 3, 4}},
{{1, 2, 3}, {4}},
{{1, 2}, {3, 4}},
{{1, 2}, {3}, {4}},
{{1}, {2}, {3}, {4}}]
The isomorphism type generating series is given by
.
EXAMPLES:
sage: P = species.PartitionSpecies()
sage: g = P.isotype_generating_series()
sage: g.coefficients(10)
[1, 1, 2, 3, 5, 7, 11, 15, 22, 30]
EXAMPLES:
sage: P = species.PartitionSpecies()
sage: P.structures([1,2,3]).list()
[{{1, 2, 3}}, {{1, 3}, {2}}, {{1, 2}, {3}}, {{2, 3}, {1}}, {{1}, {2}, {3}}]