AlternatingSignMatrices

The $n \times n$ alternating sign matrices — entries in $\{-1, 0, 1\}$, every row and column summing to 1 with alternating nonzero signs — as a lazy indexed family.

WikipediaAlternating sign matrix↗MathWorldAlternatingSignMatrix↗WikidataQ3848436↗OEISA005130↗SageMathAlternatingSignMatrices(n)↗

AlternatingSignMatrices(size)the ASMs of size size × size.

Typed signatures

Type	Package	Applies
(integer<0..>) -> indexed_collection<list<list<integer>>>	combinatorics	always

Domain: Collections

Details

• A lazy indexed collection; the count is the ASM number $A(n) = \prod_{j=0}^{n-1} \frac{(3j+1)!}{(n+j)!}$ (the Robbins numbers, OEIS A005130) — $A(4) = 42$.
• Each element is the matrix's rows; every row and column sums to 1, and every partial sum reading a row or column from its start lies in $\{0, 1\}$ — the alternating-sign condition.
• The permutation matrices are exactly the ASMs with no $-1$ entry; $A(n) \ge n!$ for every $n$, with equality only at $n \le 2$.

Enumeration

Examples

1 more kept as data, checked against the oracles

Basic #

$Count(AlternatingSignMatrices(4)) = 42$, the ASM numbers A005130

live ↗

In$\mathrm{Count}(\mathrm{AlternatingSignMatrices}(4))$Out$42$

About

Examples	2
Crosswalk	WikipediaMathWorldWikidataOEISSageMath

See also: Count, At