Weighted Integer Vectors

AUTHORS:

• Mike Hansen (2007): initial version, ported from MuPAD-Combinat

• Nicolas M. Thiery (2010-10-30): WeightedIntegerVectors(weights) + cleanup

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

Bases: Parent, UniqueRepresentation

The class of integer vectors of \(n\) weighted by weight, that is, the nonnegative integer vectors \((v_1, \ldots, v_{\ell})\) satisfying \(\sum_{i=1}^{\ell} v_i w_i = n\) where \(\ell\) is length(weight) and \(w_i\) is weight[i].

INPUT:

• n – nonnegative integer (optional)

• weight – tuple (or list or iterable) of positive integers

EXAMPLES:

Sage

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: w = WeightedIntegerVectors(8, [1,1,2]).random_element()
sage: w.parent() is WeightedIntegerVectors(8, [1,1,2])
True

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]

Python

>>> from sage.all import *
>>> WeightedIntegerVectors(Integer(8), [Integer(1),Integer(1),Integer(2)])
Integer vectors of 8 weighted by [1, 1, 2]
>>> WeightedIntegerVectors(Integer(8), [Integer(1),Integer(1),Integer(2)]).first()
[0, 0, 4]
>>> WeightedIntegerVectors(Integer(8), [Integer(1),Integer(1),Integer(2)]).last()
[8, 0, 0]
>>> WeightedIntegerVectors(Integer(8), [Integer(1),Integer(1),Integer(2)]).cardinality()
25
>>> w = WeightedIntegerVectors(Integer(8), [Integer(1),Integer(1),Integer(2)]).random_element()
>>> w.parent() is WeightedIntegerVectors(Integer(8), [Integer(1),Integer(1),Integer(2)])
True

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

Sage Live

WeightedIntegerVectors(8, [1,1,2])
WeightedIntegerVectors(8, [1,1,2]).first()
WeightedIntegerVectors(8, [1,1,2]).last()
WeightedIntegerVectors(8, [1,1,2]).cardinality()
w = WeightedIntegerVectors(8, [1,1,2]).random_element()
w.parent() is WeightedIntegerVectors(8, [1,1,2])
WeightedIntegerVectors([1,1,2])
WeightedIntegerVectors([1,1,2]).cardinality()
WeightedIntegerVectors([1,1,2]).first()

Todo

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

Element

alias of IntegerVector

class sage.combinat.integer_vector_weighted.WeightedIntegerVectors_all(weight)

Bases: DisjointUnionEnumeratedSets

Set of weighted integer vectors.

EXAMPLES:

Sage

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]

Python

>>> from sage.all import *
>>> W = WeightedIntegerVectors([Integer(3),Integer(1),Integer(1),Integer(2),Integer(1)]); W
Integer vectors weighted by [3, 1, 1, 2, 1]
>>> W.cardinality()
+Infinity

>>> W12 = W.graded_component(Integer(12))
>>> W12.an_element()
[4, 0, 0, 0, 0]
>>> W12.last()
[0, 12, 0, 0, 0]
>>> W12.cardinality()
441
>>> 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]

Sage Live

W = WeightedIntegerVectors([3,1,1,2,1]); W
W.cardinality()
W12 = W.graded_component(12)
W12.an_element()
W12.last()
W12.cardinality()
for w in W12: print(w)

grading(x)

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> C = WeightedIntegerVectors([Integer(2),Integer(1),Integer(3)])
>>> C.grading((Integer(2),Integer(1),Integer(1)))
8

Sage Live

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

subset(size=None)

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> C = WeightedIntegerVectors([Integer(2),Integer(1),Integer(3)])
>>> C.subset(Integer(4))
Integer vectors of 4 weighted by [2, 1, 3]

Sage Live

C = WeightedIntegerVectors([2,1,3])
C.subset(4)

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

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

INPUT:

• n – integer

• l – the weights in weakly decreasing order

EXAMPLES:

Sage

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]]

Python

>>> from sage.all import *
>>> from sage.combinat.integer_vector_weighted import iterator_fast
>>> list(iterator_fast(Integer(3), [Integer(2),Integer(1),Integer(1)]))
[[1, 1, 0], [1, 0, 1], [0, 3, 0], [0, 2, 1], [0, 1, 2], [0, 0, 3]]
>>> list(iterator_fast(Integer(2), [Integer(2)]))
[[1]]

Sage Live

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

Test that Issue #20491 is fixed:

Sage

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

Python

>>> from sage.all import *
>>> type(list(iterator_fast(Integer(2), [Integer(2)]))[Integer(0)][Integer(0)])
<class 'sage.rings.integer.Integer'>

Sage Live

type(list(iterator_fast(2, [2]))[0][0])

Make live