Weighted Integer Vectors

AUTHORS:

• Mike Hansen (2007): initial version, ported from MuPAD-Combinat
• Nicolas M. Thiery (2010-10-30): WeightedIntegerVectors(weights) + cleanup

Warning

The list(self) function in this file used the Permutation class improperly, returning a list of, generally speaking, invalid permutations (repeated entries, including 0).

sage.combinat.integer_vector_weighted.WeightedIntegerVectors(n=None, weight=None)

Returns the combinatorial class of integer vectors of n weighted by weight, that is, the nonnegative integer vectors \((v_1,\dots,v_{length(weight)})\) satisfying \(\sum_i v_i weight[i]==n\).

INPUT:

• n – a non negative integer (optional)
• weight – a tuple (or list or iterable) of positive integers

EXAMPLES:

sage: WeightedIntegerVectors(8, [1,1,2])
Integer vectors of 8 weighted by [1, 1, 2]
sage: WeightedIntegerVectors(8, [1,1,2]).first()
[0, 0, 4]
sage: WeightedIntegerVectors(8, [1,1,2]).last()
[8, 0, 0]
sage: WeightedIntegerVectors(8, [1,1,2]).cardinality()
25
sage: WeightedIntegerVectors(8, [1,1,2]).random_element()
[1, 1, 3]

sage: WeightedIntegerVectors([1,1,2])
Integer vectors weighted by [1, 1, 2]
sage: WeightedIntegerVectors([1,1,2]).cardinality()
+Infinity
sage: WeightedIntegerVectors([1,1,2]).first()
[0, 0, 0]

TESTS:

sage: WeightedIntegerVectors(None,None)
Traceback (most recent call last):
...
ValueError: weights should be specified

Todo

should the order of the arguments n and weight be exchanged to simplify the logic ?

class sage.combinat.integer_vector_weighted.WeightedIntegerVectors_all(weights)

Bases: sage.sets.disjoint_union_enumerated_sets.DisjointUnionEnumeratedSets

Set of weighted integer vectors.

EXAMPLES:

sage: W = WeightedIntegerVectors([3,1,1,2,1]); W
Integer vectors weighted by [3, 1, 1, 2, 1]
sage: W.cardinality()
+Infinity

sage: W12 = W.graded_component(12)
sage: W12.an_element()
[4, 0, 0, 0, 0]
sage: W12.last()
[0, 12, 0, 0, 0]
sage: W12.cardinality()
441
sage: for w in W12: print(w)
[4, 0, 0, 0, 0]
[3, 0, 0, 1, 1]
[3, 0, 1, 1, 0]
...
[0, 11, 1, 0, 0]
[0, 12, 0, 0, 0]

grading(x)

EXAMPLES:

sage: C = WeightedIntegerVectors([2,1,3])
sage: C.grading((2,1,1))
8

subset(size=None)

EXAMPLES:

sage: C = WeightedIntegerVectors([2,1,3])
sage: C.subset(4)
Integer vectors of 4 weighted by [2, 1, 3]

class sage.combinat.integer_vector_weighted.WeightedIntegerVectors_nweight(n, weight)

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent

TESTS:

sage: C = WeightedIntegerVectors(8, [1,1,2])
sage: C.__class__
<class 'sage.combinat.integer_vector_weighted.WeightedIntegerVectors_nweight_with_category'>
sage: TestSuite(C).run()

sage.combinat.integer_vector_weighted.iterator_fast(n, l)

Iterate over all l weighted integer vectors with total weight n.

INPUT:

• n – an integer
• l – the weights in weakly decreasing order

EXAMPLES:

sage: from sage.combinat.integer_vector_weighted import iterator_fast
sage: list(iterator_fast(3, [2,1,1]))
[[1, 1, 0], [1, 0, 1], [0, 3, 0], [0, 2, 1], [0, 1, 2], [0, 0, 3]]
sage: list(iterator_fast(2, [2]))
[[1]]

Test that trac ticket #20491 is fixed:

sage: type(list(iterator_fast(2, [2]))[0][0])
<type 'sage.rings.integer.Integer'>