Characteristic Species¶
- class sage.combinat.species.characteristic_species.CharacteristicSpecies(n, min=None, max=None, weight=None)[source]¶
- Bases: - GenericCombinatorialSpecies,- UniqueRepresentation- Return the characteristic species of order \(n\). - This species has exactly one structure on a set of size \(n\) and no structures 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()[0:4] [0, 1, 0, 0] sage: X.isotype_generating_series()[0:4] [0, 1, 0, 0] sage: X.cycle_index_series()[0:4] # needs sage.modules [0, p[1], 0, 0] sage: F = species.CharacteristicSpecies(3) sage: c = F.generating_series()[0:4] sage: F._check() True sage: F == loads(dumps(F)) True - >>> from sage.all import * >>> X = species.CharacteristicSpecies(Integer(1)) >>> X.structures([Integer(1)]).list() [1] >>> X.structures([Integer(1),Integer(2)]).list() [] >>> X.generating_series()[Integer(0):Integer(4)] [0, 1, 0, 0] >>> X.isotype_generating_series()[Integer(0):Integer(4)] [0, 1, 0, 0] >>> X.cycle_index_series()[Integer(0):Integer(4)] # needs sage.modules [0, p[1], 0, 0] >>> F = species.CharacteristicSpecies(Integer(3)) >>> c = F.generating_series()[Integer(0):Integer(4)] >>> F._check() True >>> F == loads(dumps(F)) True 
- class sage.combinat.species.characteristic_species.CharacteristicSpeciesStructure(parent, labels, list)[source]¶
- Bases: - GenericSpeciesStructure- automorphism_group()[source]¶
- Return 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() # needs sage.groups Symmetric group of order 3! as a permutation group - >>> from sage.all import * >>> F = species.CharacteristicSpecies(Integer(3)) >>> a = F.structures(["a", "b", "c"]).random_element(); a {'a', 'b', 'c'} >>> a.automorphism_group() # needs sage.groups Symmetric group of order 3! as a permutation group 
 - canonical_label()[source]¶
- 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'} - >>> from sage.all import * >>> F = species.CharacteristicSpecies(Integer(3)) >>> a = F.structures(["a", "b", "c"]).random_element(); a {'a', 'b', 'c'} >>> a.canonical_label() {'a', 'b', 'c'} 
 - transport(perm)[source]¶
- Return 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)) # needs sage.groups sage: a.transport(p) # needs sage.groups {'a', 'b', 'c'} - >>> from sage.all import * >>> F = species.CharacteristicSpecies(Integer(3)) >>> a = F.structures(["a", "b", "c"]).random_element(); a {'a', 'b', 'c'} >>> p = PermutationGroupElement((Integer(1),Integer(2))) # needs sage.groups >>> a.transport(p) # needs sage.groups {'a', 'b', 'c'} 
 
- sage.combinat.species.characteristic_species.CharacteristicSpecies_class[source]¶
- alias of - CharacteristicSpecies
- class sage.combinat.species.characteristic_species.EmptySetSpecies(min=None, max=None, weight=None)[source]¶
- Bases: - CharacteristicSpecies- Return 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()[0:4] [1, 0, 0, 0] sage: X.isotype_generating_series()[0:4] [1, 0, 0, 0] sage: X.cycle_index_series()[0:4] # needs sage.modules [p[], 0, 0, 0] - >>> from sage.all import * >>> X = species.EmptySetSpecies() >>> X.structures([]).list() [{}] >>> X.structures([Integer(1),Integer(2)]).list() [] >>> X.generating_series()[Integer(0):Integer(4)] [1, 0, 0, 0] >>> X.isotype_generating_series()[Integer(0):Integer(4)] [1, 0, 0, 0] >>> X.cycle_index_series()[Integer(0):Integer(4)] # needs sage.modules [p[], 0, 0, 0] 
- sage.combinat.species.characteristic_species.EmptySetSpecies_class[source]¶
- alias of - EmptySetSpecies
- class sage.combinat.species.characteristic_species.SingletonSpecies(min=None, max=None, weight=None)[source]¶
- Bases: - CharacteristicSpecies- Return 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()[0:4] [0, 1, 0, 0] sage: X.isotype_generating_series()[0:4] [0, 1, 0, 0] sage: X.cycle_index_series()[0:4] # needs sage.modules [0, p[1], 0, 0] - >>> from sage.all import * >>> X = species.SingletonSpecies() >>> X.structures([Integer(1)]).list() [1] >>> X.structures([Integer(1),Integer(2)]).list() [] >>> X.generating_series()[Integer(0):Integer(4)] [0, 1, 0, 0] >>> X.isotype_generating_series()[Integer(0):Integer(4)] [0, 1, 0, 0] >>> X.cycle_index_series()[Integer(0):Integer(4)] # needs sage.modules [0, p[1], 0, 0] 
- sage.combinat.species.characteristic_species.SingletonSpecies_class[source]¶
- alias of - SingletonSpecies