Composition species

class sage.combinat.species.composition_species.CompositionSpecies(F, G, min=None, max=None, weight=None)

Bases: GenericCombinatorialSpecies, UniqueRepresentation

Return the composition of two species.

EXAMPLES:

Sage

sage: E = species.SetSpecies()
sage: C = species.CycleSpecies()
sage: S = E(C)
sage: S.generating_series()[:5]
[1, 1, 1, 1, 1]
sage: E(C) is S
True

Python

>>> from sage.all import *
>>> E = species.SetSpecies()
>>> C = species.CycleSpecies()
>>> S = E(C)
>>> S.generating_series()[:Integer(5)]
[1, 1, 1, 1, 1]
>>> E(C) is S
True

weight_ring()

Return the weight ring for this species. This is determined by asking Sage’s coercion model what the result is when you multiply (and add) elements of the weight rings for each of the operands.

EXAMPLES:

Sage

sage: E = species.SetSpecies(); C = species.CycleSpecies()
sage: L = E(C)
sage: L.weight_ring()
Rational Field

Python

>>> from sage.all import *
>>> E = species.SetSpecies(); C = species.CycleSpecies()
>>> L = E(C)
>>> L.weight_ring()
Rational Field

class sage.combinat.species.composition_species.CompositionSpeciesStructure(parent, labels, pi, f, gs)

Bases: GenericSpeciesStructure

change_labels(labels)

Return a relabelled structure.

INPUT:

• labels – list of labels

OUTPUT:

A structure with the \(i\)-th label of self replaced with the \(i\)-th label of the list.

EXAMPLES:

Sage

sage: E = species.SetSpecies(); C = species.CycleSpecies()
sage: L = E(C)
sage: S = L.structures(['a','b','c']).list()                                # needs sage.libs.flint
sage: a = S[2]; a                                                           # needs sage.libs.flint
F-structure: {{'a', 'c'}, {'b'}}; G-structures: (('a', 'c'), ('b'))
sage: a.change_labels([1,2,3])                                              # needs sage.libs.flint
F-structure: {{1, 3}, {2}}; G-structures: [(1, 3), (2)]

Python

>>> from sage.all import *
>>> E = species.SetSpecies(); C = species.CycleSpecies()
>>> L = E(C)
>>> S = L.structures(['a','b','c']).list()                                # needs sage.libs.flint
>>> a = S[Integer(2)]; a                                                           # needs sage.libs.flint
F-structure: {{'a', 'c'}, {'b'}}; G-structures: (('a', 'c'), ('b'))
>>> a.change_labels([Integer(1),Integer(2),Integer(3)])                                              # needs sage.libs.flint
F-structure: {{1, 3}, {2}}; G-structures: [(1, 3), (2)]

transport(perm)

EXAMPLES:

Sage

sage: p = PermutationGroupElement((2,3))                                    # needs sage.groups
sage: E = species.SetSpecies(); C = species.CycleSpecies()
sage: L = E(C)
sage: S = L.structures(['a','b','c']).list()                                # needs sage.libs.flint
sage: a = S[2]; a                                                           # needs sage.libs.flint
F-structure: {{'a', 'c'}, {'b'}}; G-structures: (('a', 'c'), ('b'))
sage: a.transport(p)                                                        # needs sage.groups sage.libs.flint
F-structure: {{'a', 'b'}, {'c'}}; G-structures: (('a', 'c'), ('b'))

Python

>>> from sage.all import *
>>> p = PermutationGroupElement((Integer(2),Integer(3)))                                    # needs sage.groups
>>> E = species.SetSpecies(); C = species.CycleSpecies()
>>> L = E(C)
>>> S = L.structures(['a','b','c']).list()                                # needs sage.libs.flint
>>> a = S[Integer(2)]; a                                                           # needs sage.libs.flint
F-structure: {{'a', 'c'}, {'b'}}; G-structures: (('a', 'c'), ('b'))
>>> a.transport(p)                                                        # needs sage.groups sage.libs.flint
F-structure: {{'a', 'b'}, {'c'}}; G-structures: (('a', 'c'), ('b'))

sage.combinat.species.composition_species.CompositionSpecies_class

alias of CompositionSpecies