TITLE: Counting 2m X 2m 0-1 matrices with m ones in each row and each column.
QUESTION [14 upvotes]: Given $m>1$, what is the number of $2m\times 2m$ matrices, made of $0$ and $1$, such that each row has exactly $m$ ones, and each column has exactly $m$ zeros.
I am not sure if this is a well-known problem.

REPLY [25 votes]: An explicit formula for this was published about 30 years ago, but it was wrong. As the matter stands, there is no explicit formula.  The values up to m=15 are here.  The value for m=16 is known too, let me know if you'd like me to track it down.  The asymptotic value appears in this paper.  If I'm not mis-translating it is
$$ e^{-1/2+o(1)}\frac{\binom{2m}{m}^{4m}}{\binom{4m^2}{2m^2}}, $$
which you could apply Stirling's formula to.