Suppose $M_{n}^{k}$ is the number of regular matrices in $M_n(\mathbb{F}_2)$, that have exactly $k$ non-zero entries. Is there some sort of formula to calculate $M_n^k$?
$$(k < n\;\lor\;k > n^2 - n + 1)\overset{\text{pigeonhole principle}}\implies M_n^k = 0$$
(in $1^{\text{st}}$ case we always have at least one zero row, in $2^{\text{nd}}$ case we always have at least two identical rows). If $k = n$, then all such regular matrices have to be permutation matrices. Thus $M_n^n = n!$. However, I do not know, how to deal with the situation, where $n < k < n^2 - n + 1$.
Any help will be appreciated.