Gerechte Designs with Rectangular Regions

TL;DR

This paper studies rectangular gerechte frameworks, which are partitions of an n-by-n grid into regions of n cells, and investigates conditions under which these frameworks can be realized as Latin squares.

Contribution

It proves that all gerechte frameworks with regions as s×t or t×s rectangles are realizable as Latin squares.

Findings

All frameworks with rectangular regions of size s×t or t×s are realizable.

Progress towards understanding whether all rectangular gerechte frameworks have realizations.

Provides partial results supporting the plausibility of realizability for rectangular frameworks.

Abstract

A \emph{gerechte framework} is a partition of an $n \times n$ array into $n$ regions of $n$ cells each. A \emph{realization} of a gerechte framework is a latin square of order $n$ with the property that when its cells are partitioned by the framework, each region contains exactly one copy of each symbol. A \emph{gerechte design} is a gerechte framework together with a realization. We investigate gerechte frameworks where each region is a rectangle. It seems plausible that all such frameworks have realizations, and we present some progress towards answering this question. In particular, we show that for all positive integers $s$ and $t$, any gerechte framework where each region is either an $s \times t$ rectangle or a $t\times s$ rectangle is realizable.