Multi-latin squares

Abstract

A multi-latin square of order n and index k is an n × n array of multisets, each of cardinality k , such that each symbol from a fixed set of size n occurs k times in each row and k times in each column. A multi-latin square of index k is also referred to as a k -latin square. A 1 -latin square is equivalent to a latin square, so a multi-latin square can be thought of as a generalization of a latin square. In this note we show that any partially filled-in k -latin square of order m embeds in a k -latin square of order n , for each n ≥ 2 m , thus generalizing Evans’ Theorem. Exploiting this result, we show that there exist non-separable k -latin squares of order n for each n ≥ k + 2 . We also show that for each n ≥ 1 , there exists some finite value g ( n ) such that for all k ≥ g ( n ) , every k -latin square of order n is separable. We discuss the connection between k -latin squares and related combinatorial objects such as orthogonal arrays, latin parallelepipeds, semi-latin squares and k -latin trades. We also enumerate and classify k -latin squares of small orders.