  Let $L$ be an $n\times n$ array whose top left $r\times r$ subarray is filled
with $k$ different symbols, each occurring at most once in each row and at most
once in each column. We establish 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, $L$ is symmetric with
respect to the main diagonal, 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 Cruse (J. Combin. Theory Ser. A 16 (1974), 18-22), and the
case where the top left subarray is $r\times n$ and the symmetry is not
required, was settled by Goldwasser et al. (J. Combin. Theory Ser. A 130
(2015), 26-41). Our result allows the entries of the main diagonal to be
specified as well, which leads to an extension of the Andersen-Hoffman's
Theorem (Annals of Disc. Math. 15 (1982) 9-26, European J. Combin. 4 (1983)
33-35).
