# Alternating Sign Matrices and Hypermatrices, and a Generalization of Latin Square

## TL;DR

This paper introduces alternating sign hypermatrices (ASHMs) as a generalization of alternating sign matrices (ASMs), explores their properties, maximum nonzeros, relations to Latin squares, and addresses hypermatrix completion problems.

## Contribution

It extends the concept of ASMs to hypermatrices, establishes key properties, and investigates completion problems, providing new insights into higher-dimensional combinatorial structures.

## Key findings

- Maximum nonzeros in n×n×n ASHMs determined
- Connections between ASHMs and Latin squares established
- Several theorems on hypermatrix completion proved

## Abstract

An alternating sign matrix, or ASM, is a $(0, \pm 1)$-matrix where the nonzero entries in each row and column alternate in sign. We generalize this notion to hypermatrices: an $n\times n\times n$ hypermatrix $A=[a_{ijk}]$ is an {\em alternating sign hypermatrix}, or ASHM, if each of its planes, obtained by fixing one of the three indices, is an ASM. Several results concerning ASHMs are shown, such as finding the maximum number of nonzeros of an $n\times n\times n$ ASHM, and properties related to Latin squares. Moreover, we investigate completion problems, in which one asks if a subhypermatrix can be completed (extended) into an ASHM. We show several theorems of this type.