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Restricted r-Stirling numbers

{nk}≤m,r\left\{ {n \atop k} \right\}_{\leq m, r} is the number of partitions of an (n+r)(n + r) element set into (k+r)(k + r) subsets, such that rr distinguished elements have to be in distinct blocks, and each block has at most mm blocks.

Recurrence

{nk}≤m,r=∑i=0m−1(n−1i){n−i−1k−1}≤m,r+r∑i=0m−2(n−1i){n−i−1k}≤m,r−1.\left\{ {n \atop k} \right\}_{\leq m, r} = \sum_{i=0}^{m-1} \binom{n-1}{i} \left\{ {n-i-1 \atop k-1} \right\}_{\leq m, r} + r \sum_{i=0}^{m-2} \binom{n-1}{i} \left\{ {n-i-1 \atop k} \right\}_{\leq m, r-1}. {nk}≤m,r={n−1k−1}≤m,r+(k+r){n−1k}≤m,r−r(n−1m−1){n−mk}≤m,r−1−(n−1m){n−1−mk−1}≤m,r.\left\{ {n \atop k} \right\}_{\leq m, r} = \left\{ {n-1 \atop k-1} \right\}_{\leq m, r} + (k + r) \left\{ {n-1 \atop k} \right\}_{\leq m, r} - r \binom{n-1}{m-1} \left\{ {n-m \atop k} \right\}_{\leq m, r-1} - \binom{n-1}{m} \left\{ {n-1-m \atop k-1} \right\}_{\leq m, r}.

Generating Functions

∑n=kkm+(m−1)r{nk}≤m,rtnn!=1k!(1+t+t22!+⋯+tm−1(m−1)!)r(t+t22!+⋯+tmm!)k.\sum_{n=k}^{km+(m-1)r} \left\{ {n \atop k} \right\}_{\leq m, r} \frac{t^n}{n!} = \frac{1}{k!} \left( 1 + t + \frac{t^2}{2!} + \cdots + \frac{t^{m-1}}{(m-1)!} \right)^r \left( t + \frac{t^2}{2!} + \cdots + \frac{t^m}{m!} \right)^k.

References

Komatsu and Ramirez: Generalized poly-Cauchy and poly-Bernoulli numbers by using incomplete r-Stirling numbers

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