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)
__init__(parent, labels, list)

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
__repr__()

EXAMPLES:

sage: S = species.PartitionSpecies()
sage: a = S.structures(["a","b","c"]).random_element(); a
{{'a', 'b', 'c'}}
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)
__init__(min=None, max=None, weight=None)

EXAMPLES:

sage: P = species.PartitionSpecies()
sage: P._check()
True
sage: P == loads(dumps(P))
True
static _cached_constructor(*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]
_canonical_rep_from_partition(structure_class, labels, p)

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}}
_cis(series_ring, base_ring)

The cycle index series for the species of partitions is given by

exp \sum_{n \ge 1} \frac{1}{n} \left( exp \left( \sum_{k \ge 1} \frac{x_{kn}}{k} \right) -1 \right).

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]]
_default_structure_class
alias of PartitionSpeciesStructure
_gs_iterator(base_ring)

EXAMPLES:

sage: P = species.PartitionSpecies()
sage: g = P.generating_series()
sage: g.coefficients(5)
[1, 1, 1, 5/6, 5/8]
_isotypes(structure_class, labels)

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}}]
_itgs_iterator(base_ring)

The isomorphism type generating series is given by \frac{1}{1-x}.

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]
_structures(structure_class, labels)

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}}]

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