# Counting 0-1 $n\times n$ matrices with a given rank r

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.

• The Zivkovic reference there is the latest I know. Considering the large number of such matrices, I suspect you will not see an exact and useful enumeration any time soon. I suspect even approximations will be very rough. Gerhard "Doesn't Know The Rank Asymptotics" Paseman, 2017.10.10. Oct 10, 2017 at 18:55
• Possible duplicate of Number rank-k 0-1 matrices (characteristic 0) Oct 10, 2017 at 21:10
• @777, this question asks for estimates, that one doesn't. Not quite a duplicate, I think. Oct 10, 2017 at 22:17
• @GerryMyerson mathoverflow.net/questions/195326/… Oct 10, 2017 at 22:58
• Possible duplicate of Enumerating matrices function of ranks Oct 10, 2017 at 23:10