矩形区域的Gerechte设计
Gerechte Designs with Rectangular Regions
摘要：
\emph{gerechte framework}是将$n \times n$阵列划分成各自为$n$细胞的$n$区域的分区。Gerechte框架的\emph{realization}是一个$n$阶的拉丁方，其属性是当其单元被框架划分时，每个区域包含每个符号的一个副本。\emph{gerechte design}是一个带有实现的通用框架。我们调查gerechte框架，每个区域是一个矩形。似乎所有这些框架都有实现，我们在回答这个问题方面取得了一些进展。特别是，我们表明，对于所有正整数$s$和$t$，任何gerechte框架，其中每个区域是一个$s \times t$矩形或一个$t\times s$矩形是可实现的。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
