I would like to find an explicit, analytical formula that counts the number of $n\times m$ matrices with given sums $c_1, \ldots , c_n$ along the columns and only one non-zero value along each row (but the rows' sums may not be specified).
Do you think an explicit analytical solution can be found?
I need an analytical formula because I then would like to see what a "typical matrix" of this set looks like.
NOTE : The problem can also be thought as a multiset partition problem with constrains. I have a multiset with multiplicities $c_1, \ldots , c_n$ and I want to see what a typical partition looks like when we force the block $i$ of a partition to be a multiset with multiplicity $d_i$ (so only one species for each block of each possible partition).