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]#
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() # 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
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()[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
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()[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