Let L be an n x n array whose top left r x 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), and the case s=n was settled by Goldwasser et al. (2015). Our technique leads to a very short proof of the latter.
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