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.



Counting 2m X 2m 0-1 matrices with m ones in each row and each column.
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](http://cs.anu.edu.au/~bdm/data/semiregular.html). 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](http://www.combinatorics.org/ojs/index.php/eljc/article/view/v12i1r29). 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.


