Sum species¶

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

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)
True
sage: F._check()
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)
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)

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']).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