What is the number $N$ of $n \times n$ $0$-$1$ matrices with rank $k$?
I read this sequence is "OEIS A064230 Triangle $T(n,k)$ = number of rational (0,1) matrices of rank $k$ ($n\ge 0$, $0\le k\le n$)".
I am interested in approximation results, i.e. finding upper and/or lower bound for $N$ when $k=\Theta(f(n))$ for some function $f$.
Thank you! PS: I would like to emphasize that, for this "counting problem" context, I am asking only for (even rough) estimates.