Consider a square matrix of size N, such that each row and each column has exactly M non-zero positive elements. A simple example with N=3 and M=2 would be
$ \left( \begin{array}{ccc} 0 & A1 & A2 \\ A3 & 0 & A4 \\ A5 & A6 & 0 \\ \end{array} \right)$
Notice that this matrix has rank 3 (i.e. the maximum) In general, for a given M, can one calculate the value of N that ensures that the corresponding matrix has the maximum rank? (i.e., rank = N)
Based on some tests that I did in a different context (too long to be reported here) I believe that the relation is $N = \frac{(2M-1)!}{M! (M-1)!}$ but I wonder if this can be proved rigorously.