Partition Species¶
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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
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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
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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)]
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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'}}]
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change_labels
(labels)¶ Return a relabelled structure.
INPUT:
labels
, a list of labels.
OUTPUT:
A structure with the i-th label of self replaced with the i-th label of the list.
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}}
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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}}
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sage.combinat.species.partition_species.
PartitionSpecies_class
¶ alias of
sage.combinat.species.partition_species.PartitionSpecies