I am looking for a paper or guidance for expanding the r-associated Stirling numbers of the second kind $S_r(n,k)$. $S_r(n,k)$ is the number of ways to partition a set of n objects into k subsets, with each subset containing at least r elements. I want to calculate $S_r(n,k,b)$ which also counts the number of times b that a subset contains exactly r elements. It is obvious that $S_r(n,k)=\sum_{b=0}^kS_r(n,k,b)$ but can I get an analytic expression for $S_r(n,k,b)$? I think it is possible to solve this with generating functions but I can use some guidance.
2026-03-25 23:38:27.1774481907
Expansion for r-associated Stirling numbers of the second kind
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