Multi-latin squares

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. Hämäläinen supported by the Eduard Cech Center under the grant LC505. Lefevre supported by grants DP0664030 and LX0453416. Supported by ARC Grant DP0662946 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 ≥ 2m, thus generalizing Evans’ Theorem. Exploiting this result, we show that there exist nonseparable 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.