论文标题
Ryser的定理，用于$ρ$ -LATIN矩形
Ryser's Theorem for $ρ$-latin Rectangles
论文作者
论文摘要
令$ l $为$ n \ times n $阵列，其顶部剩下的$ r \ times s $子阵列充满了$ k $不同的符号，每个符号最多一次发生在每一行，每列最多一次。我们发现必要且充分的条件，以确保$ L $的其余单元可以填充，以使每个符号最多一次在每一行，最多一次，每列最多一次，并且每个符号在$ L $中发生一个规定的次数。每个符号发生的规定次数为$ n $的情况是由Ryser（Proc。Amer。Math。Soc。2（1951），550--552）解决的，Case $ S = N $由Goldwasser等人解决。 （J. Combin。理论Ser。A130（2015），26--41）。我们的技术导致了后者的简短证明。
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 (Proc. Amer. Math. Soc. 2 (1951), 550--552), and the case $s=n$ was settled by Goldwasser et al. (J. Combin. Theory Ser. A 130 (2015), 26--41). Our technique leads to a very short proof of the latter.
