Product species#

class sage.combinat.species.product_species.ProductSpecies(F, G, min=None, max=None, weight=None)#

Bases: GenericCombinatorialSpecies, UniqueRepresentation

EXAMPLES:

sage: X = species.SingletonSpecies()
sage: A = X*X
sage: A.generating_series()[0:4]
[0, 0, 1, 0]

sage: P = species.PermutationSpecies()
sage: F = P * P; F
Product of (Permutation species) and (Permutation species)
sage: F == loads(dumps(F))
True
sage: F._check()                                                            # needs sage.libs.flint
True

left_factor()#

Returns the left factor of this product.

EXAMPLES:

sage: P = species.PermutationSpecies()
sage: X = species.SingletonSpecies()
sage: F = P*X
sage: F.left_factor()
Permutation species

right_factor()#

Returns the right factor of this product.

EXAMPLES:

sage: P = species.PermutationSpecies()
sage: X = species.SingletonSpecies()
sage: F = P*X
sage: F.right_factor()
Singleton species

weight_ring()#

Returns 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: S = species.SetSpecies()
sage: C = S*S
sage: C.weight_ring()
Rational Field

sage: S = species.SetSpecies(weight=QQ['t'].gen())
sage: C = S*S
sage: C.weight_ring()
Univariate Polynomial Ring in t over Rational Field

sage: S = species.SetSpecies()
sage: C = (S*S).weighted(QQ['t'].gen())
sage: C.weight_ring()
Univariate Polynomial Ring in t over Rational Field

class sage.combinat.species.product_species.ProductSpeciesStructure(parent, labels, subset, left, right)#

Bases: GenericSpeciesStructure

automorphism_group()#

EXAMPLES:

sage: # needs sage.groups
sage: p = PermutationGroupElement((2,3))
sage: S = species.SetSpecies()
sage: F = S * S
sage: a = F.structures([1,2,3,4])[1]; a
{1}*{2, 3, 4}
sage: a.automorphism_group()
Permutation Group with generators [(2,3), (2,3,4)]

sage: [a.transport(g) for g in a.automorphism_group()]                      # needs sage.groups
[{1}*{2, 3, 4},
 {1}*{2, 3, 4},
 {1}*{2, 3, 4},
 {1}*{2, 3, 4},
 {1}*{2, 3, 4},
 {1}*{2, 3, 4}]

sage: a = F.structures([1,2,3,4])[8]; a                                     # needs sage.groups
{2, 3}*{1, 4}
sage: [a.transport(g) for g in a.automorphism_group()]                      # needs sage.groups
[{2, 3}*{1, 4}, {2, 3}*{1, 4}, {2, 3}*{1, 4}, {2, 3}*{1, 4}]

canonical_label()#

EXAMPLES:

sage: S = species.SetSpecies()
sage: F = S * S
sage: S = F.structures(['a','b','c']).list(); S
[{}*{'a', 'b', 'c'},
 {'a'}*{'b', 'c'},
 {'b'}*{'a', 'c'},
 {'c'}*{'a', 'b'},
 {'a', 'b'}*{'c'},
 {'a', 'c'}*{'b'},
 {'b', 'c'}*{'a'},
 {'a', 'b', 'c'}*{}]

sage: F.isotypes(['a','b','c']).cardinality()
4
sage: [s.canonical_label() for s in S]
[{}*{'a', 'b', 'c'},
 {'a'}*{'b', 'c'},
 {'a'}*{'b', 'c'},
 {'a'}*{'b', 'c'},
 {'a', 'b'}*{'c'},
 {'a', 'b'}*{'c'},
 {'a', 'b'}*{'c'},
 {'a', 'b', 'c'}*{}]

change_labels(labels)#

Return a relabelled structure.

INPUT:

labels, a list of labels.

OUTPUT:

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

EXAMPLES:

sage: S = species.SetSpecies()
sage: F = S * S
sage: a = F.structures(['a','b','c'])[0]; a
{}*{'a', 'b', 'c'}
sage: a.change_labels([1,2,3])
{}*{1, 2, 3}

transport(perm)#

EXAMPLES:

sage: # needs sage.groups
sage: p = PermutationGroupElement((2,3))
sage: S = species.SetSpecies()
sage: F = S * S
sage: a = F.structures(['a','b','c'])[4]; a
{'a', 'b'}*{'c'}
sage: a.transport(p)
{'a', 'c'}*{'b'}

sage.combinat.species.product_species.ProductSpecies_class#: alias of ProductSpecies