Sum species

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

Bases: sage.combinat.species.structure.GenericSpeciesStructure

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'}

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

Bases: sage.combinat.species.species.GenericCombinatorialSpecies

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