Ryser's Theorem for $\rho$-latin Rectangles

TL;DR

This paper extends Ryser's theorem to $ ho$-latin rectangles, providing necessary and sufficient conditions for completing partially filled arrays with prescribed symbol frequencies, generalizing previous results for specific cases.

Contribution

It introduces a unified set of conditions for completing $n imes n$ arrays with prescribed symbol counts, generalizing earlier theorems for special cases.

Findings

Established necessary and sufficient conditions for $ ho$-latin rectangle completion.

Provided a concise proof of Goldwasser et al.'s result for the case $s=n$.

Extended Ryser's theorem to more general array configurations.

Abstract

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.