← Back to the home page

Associated r-Stirling number

{nk}≥m,r\left\{ {n \atop k} \right\}_{\geq 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 least mm blocks.

Recurrence

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

Generating Function

∑n=mk+(m−1)r∞{nk}≥m,rtnn!=1k!(et−∑i=0m−2tii!)r(et−∑i=0m−1tii!)k\sum_{n=mk+(m-1)r}^\infty \left\{ {n \atop k} \right\}_{\geq m, r} \frac{t^n}{n!} = \frac{1}{k!} \left( e^t - \sum_{i=0}^{m-2} \frac{t^i}{i!} \right)^r \left( e^t - \sum_{i=0}^{m-1} \frac{t^i}{i!} \right)^k

References

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

Comments

Loading comments...