Gerechte designs with rectangular regions

Abstract

AbstractA gerechte framework is a partition of an n × n array into n regions of n cells each. A 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 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 toward 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 × t rectangle or a t × s rectangle is realizable. © 2011 Wiley Periodicals, Inc. J Combin Designs 20:112‐123, 2012