Let L be an $n\times n$ array whose top left $r\times s$ subarray is filled with k different symbols, each occurring at most once in each row and at most once in each column. We find necessary and sufficient conditions that ensure the remaining cells of L can be filled such that each symbol occurs at most once in each row and at most once in each column, and each symbol occurs a prescribed number of times in L. The case where the prescribed number of times each symbol occurs is n was solved by Ryser (1951) [11], and the case $s=n$ was settled by Goldwasser et al. (2015) [5]. Our technique leads to a very short proof of the latter.

中文翻译：

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ρ-拉丁矩形的 Ryser 定理

设L为$n\times n$左上角的数组$r\times s$子数组用k个不同的符号填充，每个符号在每行中最多出现一次，在每列中最多出现一次。我们找到了确保L的剩余单元格可以被填充的充分必要条件，使得每个符号在每行中最多出现一次，在每列中最多出现一次，并且每个符号在L中出现规定的次数。Ryser (1951) [11] 解决了每个符号出现规定次数为n的情况，并且该情况$s=n$由 Goldwasser 等人解决。(2015) [5]。我们的技术导致了后者的一个非常简短的证明。