Multi-latin squares

TL;DR

This paper generalizes Latin squares to multi-latin squares, proves embedding theorems, explores their properties such as separability, and connects them to other combinatorial objects, including enumeration for small cases.

Contribution

It introduces multi-latin squares, proves embedding results, demonstrates existence of non-separable cases, and establishes conditions for separability based on the index.

Findings

Any partially filled $k$-latin square can be embedded in a larger one.

Existence of non-separable $k$-latin squares for certain orders.

For large enough $k$, all $k$-latin squares of a given order are separable.

Abstract

A multi-latin square of order $n$ and index $k$ is an $n\times n$ 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 $n\geq 2m$, thus generalizing Evans' Theorem. Exploiting this result, we show that there exist non-separable $k$-latin squares of order $n$ for each $n\geq k+2$. We also show that for each $n\geq 1$, there exists some finite value $g(n)$ such that for all $k\geq g(n)$, 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.