On Completing Partial Latin Squares with Prescribed Diagonals

Necessary and sufficient numerical conditions are known for the embedding of an incomplete latin square $L$ of order $n$ into a latin square $T$ of order $t \geq 2n+1$ in which each symbol is prescribed to occur in a given number of cells on the diagonal of $T$ outside of $L$. This includes the classic case where $T$ is required to be idempotent. If $t