Consider the standard balls into bins problem, with $n$ balls uniformly randomly thrown into $k$ bins.
What is the probability that exactly $b$ bins have at least $m$ balls?
This is a generalization of the standard problem where we want to know the probability that exactly $b$ bins have at least $1$ ball (non-empty) – I know that you can use stirling numbers of the second kind to derive a closed form solution for this. Is it possible to generalize this to the event of having at least $m$ balls?
Best Answer
Hint/Approach: A way to extend the definition of Stirling numbers are the "Associated Stirling numbers of the second kind", they are denoted by ${ n\brace k}_{\geq m},$ where this means the number of partitions of $[n]$ into $k$ blocks, each one having at least $m$ elements. So, most likely if you know that the formula involves taking Stirling numbers, you can change that for these associated. You can easily check that they satisfy the following recursion $${n+1\brace k}_{\geq m} = \sum _{j = m-1}^n\binom{n}{j}{n-j\brace k-1}_{\geq m},$$ by checking the elements that are in the same block as the element $n+1.$