Riordan and Stein, in "Arrangements on Chessboards" (*Journal of Combinatorial Theory, Series A*, **12** 72-80, 1972) consider the numbers $A(r,s,k)$ defined by
$$\sum_{r,s} \binom{n}{r} \binom{m}{s} A(r,s,k) = \binom{nm}{k},$$ or, as others have pointed out, the number of $r \times s$ $(0,1)$-matrices with $k$ $1$'s and with at least one $1$ in every row and every column. They obtain two formulas for these numbers:

$$A(r,s,k) = \sum_{i,j} (-1)^{i+j+r+s} \binom{r}{i} \binom{s}{j} \binom{ij}{k},$$

$$A(r,s,k) = \sum_n \frac{r! s!}{k!} S(n,r) S(n,s) s(k,n),$$
where $S(n,r)$ and $s(k,n)$ are Stirling numbers of the second and first kinds, respectively.

They also obtain the recurrence relation
$$(k+1)A(r,s,k+1) +k A(r,s,k) $$
$$= rs \left(A(r,s,k) + A(r-1,s,k) + A(r,s-1,k) + A(r-1,s-1,k)\right)$$
and that $A(r,s,k)$ is the coefficient of $t^k$ in
$$\sum_j (-1)^{s+j} \binom{s}{j} \left( (1+t)^j-1) \right)^r,$$
as in Richard Stanley's answer.