Alternating Sign Hypermatrix Decompositions of Latin-like Squares

Abstract: To any $n \times n$ Latin square $L$, we may associate a unique sequence of mutually orthogonal permutation matrices $P = P_1, P_2, ..., P_n$ such that $L = L(P) = \sum kP_k$. 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 $n \times n \times n$ (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 $n \times n \times n$ ASHM, this generalisation of Latin squares follows very naturally, with an ASHM $A$ having corresponding ASHL $L = L(A) =\sum kA_k$, where $A_k$ is the $k<sup>{\text{th}}$ plane of $A$. This paper addresses some 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 providing a tight lower bound on $n$ for which two $n \times n \times n$ ASHMs can correspond to the same ASHL, and secondly by exploring the maximum number of times a particular integer may occur as an entry of an $n \times n$ ASHL. A general construction is given for an $n \times n$ ASHL with the same entry occurring $\lfloor\frac{n² + 4n -19}{2}\rfloor$ times, improving considerably on the previous best construction, which achieved the same entry occuring $2n$ times.