Sum species

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

Bases: sage.combinat.species.species.GenericCombinatorialSpecies, sage.structure.unique_representation.UniqueRepresentation

EXAMPLES:

sage: X = species.SingletonSpecies()
sage: A = X*X
sage: A.generating_series().coefficients(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()
True

TESTS:

sage: X = species.SingletonSpecies()
sage: X*X is X*X
True
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: sage.combinat.species.structure.GenericSpeciesStructure

TESTS:

sage: S = species.SetSpecies()
sage: F = S * S
sage: a = F.structures(['a','b','c']).random_element()
sage: a == loads(dumps(a))
True
automorphism_group()

EXAMPLES:

sage: p = PermutationGroupElement((2,3))
sage: S = species.SetSpecies()
sage: F = S * S
sage: a = F.structures([1,2,3,4]).random_element(); 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()]
[{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]).random_element(); a
{2, 3}*{1, 4}
sage: [a.transport(g) for g in a.automorphism_group()]
[{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)

EXAMPLES:

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

EXAMPLES:

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

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