Characteristic Species

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
class sage.combinat.species.characteristic_species.CharacteristicSpeciesStructure(parent, labels, list)

Bases: sage.combinat.species.structure.GenericSpeciesStructure

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
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
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'}
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'}
sage.combinat.species.characteristic_species.CharacteristicSpecies_class

alias of CharacteristicSpecies

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
sage.combinat.species.characteristic_species.EmptySetSpecies_class

alias of EmptySetSpecies

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
sage.combinat.species.characteristic_species.SingletonSpecies_class

alias of SingletonSpecies

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