Sum species¶
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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
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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
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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
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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
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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
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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}]
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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'}*{}]
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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}
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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'}
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sage.combinat.species.product_species.
ProductSpecies_class
¶ alias of
ProductSpecies