Associated Stirling number of the second kind

$$\left\{\begin{matrix}n\\k\end{matrix}\right\}_{\ge m}$$ is the number of ways of partitioning an $n$ element set into $k$ subsets such that each subset has at least $m$ elements.

Cases

• Case $m=2$: https://oeis.org/A008299
• Case $m=3$: https://oeis.org/A059022
• Case $m=4$: https://oeis.org/A059023
• Case $m=5$: https://oeis.org/A059024
• Case $m=6$: https://oeis.org/A059025

Recurrence

• $$\left\{\begin{matrix}n\\0\end{matrix}\right\}_{\ge m} = \left\{\begin{matrix}0\\n\end{matrix}\right\}_{\ge m} = 0 \text{ for } n > 0$$
• $$\left\{\begin{matrix}0\\0\end{matrix}\right\}_{\ge m} = 1$$
• $$\begin{align*} \left\{ {n+1 \atop k} \right\}_{\geq m} &= \sum_{i=m-1}^n \binom{n}{i} \left\{ {n-i \atop k-1} \right\}_{\geq m} \\ &= k \left\{ {n \atop k} \right\}_{\geq m} + \binom{n}{m-1} \left\{ {n-m+1 \atop k-1} \right\}_{\geq m} \end{align*}$$

Generating Function

$$\sum_{n=mk}^{\infty} \left\{ {n \atop k} \right\}_{\geq m} \frac{x^n}{n!} = \frac{1}{k!} \left( e^x - E_{m-1}(x) \right)^k$$

$$E_m(t) = \sum_{k=0}^{m} \frac{t^k}{k!}$$

Sources

• Komatsu, Liptai, Mező: Incomplete poly-Bernoulli numbers associated with incomplete Stirling numbers
• L. Comtet: Advanced Combinatorics, Reidel, 1974, p. 222