I searched on Google and I found something like my question in physics.harvard.edu/uploads/files/undergrad/probweek/sol38.p‌​df and in mathworld.wolfram.com/UniformSumDistribution.html, but I don't have enough knowledges to adapt them. And, regarding your last question, I am sure that I'm not interrested to aproximate it when m is close to 0. Thank you again!

Thank you very much Douglas for your interrest! You're right about my method of generation: I take all the numbers from 1 to n and I put each in one of k bins. The $n_1,...,n_k$ are the count of values placed in each bin. It's a part of a graph algorithm.

Sorry for my mistake. I am programmer and I have only few knowledges about probabilities. I think that the distribution of ${n_1,...,n_k}$ is an uniform one as I generate them using a good PRNG, isn't?

I have no specific distribution on the multiset ${n_1,...,n_k}$, but, if is mandatory in order to calculate the probability, you can assume one. Thank you!