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]
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
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
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
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'}
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'}
Bases: sage.combinat.species.species.GenericCombinatorialSpecies
EXAMPLES:
sage: F = species.CharacteristicSpecies(3)
sage: c = F.generating_series().coefficients(4)
sage: F._check()
True
sage: F == loads(dumps(F))
True
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
Bases: sage.combinat.species.characteristic_species.CharacteristicSpecies_class
EXAMPLES:
sage: E = species.EmptySetSpecies()
sage: E._check()
True
sage: E == loads(dumps(E))
True
Returns the species of singletons. This species has exactly one structure on a set of size n. 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
Bases: sage.combinat.species.characteristic_species.CharacteristicSpecies_class
EXAMPLES:
sage: S = species.SingletonSpecies()
sage: S._check()
True
sage: S == loads(dumps(S))
True