As a part of a greater question, I stumbled upon the following problem:
Let's suppose we have $n$ balls and $k$ bins. Bins are numbered $b_1, b_2, b_3,...b_k$.
Let $N[b_i]$ be the number of balls inside the $i$th bin. In how many ways can I place $n$ balls inside the $k$ bins such as $N[b_1] \geq N[b_2] \geq N[b_3] \geq ... \geq N[b_k]$?
As an example, if we have 4 balls and 3 bins, the possibilities are:
[4,0,0], [3,1,0], [2,2,0], [2,1,1].
Every help is much appreciated.