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

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

Python

>>> 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

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

Python

>>> 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

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'}

Python

>>> 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

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'}

Python

>>> 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

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]

Python

>>> 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

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]

Python

>>> 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