Alternating sign hypermatrix decompositions of Latin-like squares

Abstract

To any n × n Latin square L, we may associate a unique sequence of mutually orthogonal permutation matrices P = P_1, P_2, ..., P_n such that L = L(P ) = ∑ k_Pk . Brualdi and Dahl (2018) described a generalisation of a Latin square, called an alternating sign hypermatrix Latin-like square (ASHL), by replacing P with an alternating sign hypermatrix (ASHM). An ASHM is an n × n × n (0,1,-1)-hypermatrix in which the non-zero elements in each row, column, and vertical line alternate in sign, beginning and ending with 1. Since every sequence of n mutually orthogonal permutation matrices forms the planes of a unique n × n × n ASHM, this generalisation of Latin squares follows very naturally, with an ASHM A having corresponding ASHL L = L(A) = ∑ kA_k , where A_k is the kth plane of A. This paper addresses open problems posed in Brualdi and Dahl’s article, firstly by characterising how pairs of ASHMs with the same corresponding ASHL relate to one another and identifying the smallest dimension for which this can happen, and secondly by exploring the maximum number of times a particular integer may occur as an entry of an n × n ASHL. A construction is given for an n × n ASHL with the same entry occurring (n^2 +4n−19)/2 times, improving on the previous best of 2n.