Count Arrays

Consider all matrices with $N$ rows (numbered $1$ through $N$) and $M$ columns (numbered $1$ through $M$) containing only integers between $0$ and $K-1$ (inclusive). For each such matrix $A$, let's form a sequence $L_1, L_2, \ldots, L_{N+M}$: - For each $i$ ($1 \le i \le N$), $L_i$ is the maximum of all elements in the $i$-th row of $A$. - For each $i$ ($1 \le i \le M$), $L_{N+i}$ is the maximum