Characteristic Species¶
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class
sage.combinat.species.characteristic_species.
CharacteristicSpecies
(n, min=None, max=None, weight=None)¶ Bases:
sage.combinat.species.species.GenericCombinatorialSpecies
,sage.structure.unique_representation.UniqueRepresentation
Returns the characteristic species of order \(n\).
This species has exactly one structure on a set of of size \(n\) and no structures of on sets of any other size.
EXAMPLES:
sage: X = species.CharacteristicSpecies(1) sage: X.structures([1]).list() [1] sage: X.structures([1,2]).list() [] sage: X.generating_series().coefficients(4) [0, 1, 0, 0] sage: X.isotype_generating_series().coefficients(4) [0, 1, 0, 0] sage: X.cycle_index_series().coefficients(4) [0, p[1], 0, 0] sage: F = species.CharacteristicSpecies(3) sage: c = F.generating_series().coefficients(4) sage: F._check() True sage: F == loads(dumps(F)) True
TESTS:
sage: S1 = species.CharacteristicSpecies(1) sage: S2 = species.CharacteristicSpecies(1) sage: S3 = species.CharacteristicSpecies(2) sage: S4 = species.CharacteristicSpecies(2, weight=2) sage: S1 is S2 True sage: S1 == S3 False
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class
sage.combinat.species.characteristic_species.
CharacteristicSpeciesStructure
(parent, labels, list)¶ Bases:
sage.combinat.species.structure.GenericSpeciesStructure
This is a base class from which the classes for the structures inherit.
EXAMPLES:
sage: from sage.combinat.species.structure import GenericSpeciesStructure sage: a = GenericSpeciesStructure(None, [2,3,4], [1,2,3]) sage: a [2, 3, 4] sage: a.parent() is None True sage: a == loads(dumps(a)) True
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automorphism_group
()¶ Returns the group of permutations whose action on this structure leave it fixed. For the characteristic species, there is only one structure, so every permutation is in its automorphism group.
EXAMPLES:
sage: F = species.CharacteristicSpecies(3) sage: a = F.structures(["a", "b", "c"]).random_element(); a {'a', 'b', 'c'} sage: a.automorphism_group() Symmetric group of order 3! as a permutation group
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canonical_label
()¶ EXAMPLES:
sage: F = species.CharacteristicSpecies(3) sage: a = F.structures(["a", "b", "c"]).random_element(); a {'a', 'b', 'c'} sage: a.canonical_label() {'a', 'b', 'c'}
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transport
(perm)¶ Returns the transport of this structure along the permutation perm.
EXAMPLES:
sage: F = species.CharacteristicSpecies(3) sage: a = F.structures(["a", "b", "c"]).random_element(); a {'a', 'b', 'c'} sage: p = PermutationGroupElement((1,2)) sage: a.transport(p) {'a', 'b', 'c'}
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sage.combinat.species.characteristic_species.
CharacteristicSpecies_class
¶ alias of
CharacteristicSpecies
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class
sage.combinat.species.characteristic_species.
EmptySetSpecies
(min=None, max=None, weight=None)¶ Bases:
sage.combinat.species.characteristic_species.CharacteristicSpecies
Returns the empty set species.
This species has exactly one structure on the empty set. It is the same (and is implemented) as
CharacteristicSpecies(0)
.EXAMPLES:
sage: X = species.EmptySetSpecies() sage: X.structures([]).list() [{}] sage: X.structures([1,2]).list() [] sage: X.generating_series().coefficients(4) [1, 0, 0, 0] sage: X.isotype_generating_series().coefficients(4) [1, 0, 0, 0] sage: X.cycle_index_series().coefficients(4) [p[], 0, 0, 0]
TESTS:
sage: E1 = species.EmptySetSpecies() sage: E2 = species.EmptySetSpecies() sage: E1 is E2 True sage: E = species.EmptySetSpecies() sage: E._check() True sage: E == loads(dumps(E)) True
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sage.combinat.species.characteristic_species.
EmptySetSpecies_class
¶ alias of
EmptySetSpecies
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class
sage.combinat.species.characteristic_species.
SingletonSpecies
(min=None, max=None, weight=None)¶ Bases:
sage.combinat.species.characteristic_species.CharacteristicSpecies
Returns the species of singletons.
This species has exactly one structure on a set of size \(1\). It is the same (and is implemented) as
CharacteristicSpecies(1)
.EXAMPLES:
sage: X = species.SingletonSpecies() sage: X.structures([1]).list() [1] sage: X.structures([1,2]).list() [] sage: X.generating_series().coefficients(4) [0, 1, 0, 0] sage: X.isotype_generating_series().coefficients(4) [0, 1, 0, 0] sage: X.cycle_index_series().coefficients(4) [0, p[1], 0, 0]
TESTS:
sage: S1 = species.SingletonSpecies() sage: S2 = species.SingletonSpecies() sage: S1 is S2 True sage: S = species.SingletonSpecies() sage: S._check() True sage: S == loads(dumps(S)) True
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sage.combinat.species.characteristic_species.
SingletonSpecies_class
¶ alias of
SingletonSpecies