I am trying to count the number of $(0,1)$ $m \times n$ matrices that contain exactly $k$ ones distributed such that no row and no column contains only zeros (clearly, $k \ge m$ and $k \ge n$ yield interesting cases). I looked at similar questions (e.g., here), but they usually do not contain global constraints such as the total number of ones. Similarly, unlike here, this question does not have any constraints on column/row sums.
Thanks in advance for your help!
For instance, if $k=m\ge n,$ you are asking for the number of surjections from $[m]$ to $[n],$ a classical counting problem. Like the problem of counting surjections, your more general question has an answer in the form of an inclusion-exclusion formula: $$\sum_{u=0}^m\sum_{v=0}^n(-1)^{u+v}\binom mu\binom nv\binom{(m-u)(n-v)}k\tag1$$ or equivalently (setting $r=m-u,s=n-v$) $$\sum_{r=0}^m\sum_{s=0}^n(-1)^{m+n+r+s}\binom mr\binom ns\binom{rs}k.\tag2$$
Example. Consider the case $m=2,n=3,k=3.$ The answer is clearly $2^3-2=6.$ With $m=2$ and $n=3$ in formula (2), the only nonzero terms occur when $rs\ge3,$ that is, when $(r,s)$ is $(1,3)$ or $(2,2)$ or $(2,3),$ and we get $$-\binom21\binom33\binom33-\binom22\binom32\binom43+\binom22\binom33\binom63=-2-12+20=6.$$
Explanation. Let $E$ be the set of all $m\times n$ matrices of zeros and ones with exactly $k$ ones. For $i\in[m]$ let $A_i$ be the set of all matrices in $E$ with no ones in the $i^{\text{th}}$ row. For $j\in[n]$ let $B_j$ be the set of all matrices in $E$ with no ones in the $j^{\text{th}}$ column. The expression (1) comes from the inclusion-exclusion formula for $$|E\setminus(\bigcup_{i\in[m]}A_i\cup\bigcup_{j\in[n]}B_j)|\tag3$$ by simplifying it using the fact that, for $I\in\binom{[m]}u$ and $J\in\binom{[n]}v,$ $$|\bigcap_{i\in I}A_i\cap\bigcap_{j\in J}B_j|=\binom{(m-u)(n-v)}k.\tag4$$
