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(S, r)-Bell number

Bn,S,rB_{n, S, r} is the number of set partitions of an (n+r)(n+r) element set into blocks, such that each block has cardinality belonging to some set SS, and there are rr distinguished elements in separate blocks.

Formulas

Bn,S,r=∑k=0n{nk}S,rB_{n, S, r} = \sum_{k=0}^n \left\{ {n \atop k} \right\}_{S, r}

Recurrence

Bn,S,r+1=∑s∈S(ns−1)Bn−s+1,S,rB_{n, S, r+1} = \sum_{s \in S} \binom{n}{s-1} B_{n-s+1, S, r} (n+r)Bn,S,r=∑s∈Ss(ns)Bn−s,S,r+r∑s∈Ss(ns−1)Bn−s+1,S,r−1(n + r) B_{n, S, r} = \sum_{s \in S} s \binom{n}{s} B_{n-s, S, r} + r \sum_{s \in S} s \binom{n}{s-1} B_{n-s+1, S, r-1} Bn+1,S,r=Bn,S,r+1+r∑s∈S(ns−2)Bn−s+2,S,r−1B_{n+1, S, r} = B_{n, S, r+1} + r \sum_{s \in S} \binom{n}{s-2} B_{n-s+2, S, r-1}

Generating Function

∑n=0∞Bn,S,rxnn!=(∑i≥1xki−1(ki−1)!)rexp⁡(∑i≥1xkiki!)\sum_{n=0}^\infty B_{n, S, r} \frac{x^n}{n!} = \left( \sum_{i \geq 1} \frac{x^{k_i-1}}{(k_i-1)!} \right)^r \exp\left( \sum_{i \geq 1} \frac{x^{k_i}}{k_i!} \right)

S={k1,k2,k2,...}S = \{k_1, k_2, k_2, ...\}

References

Bényi, Méndez, Ramirez, Wakhare: RESTRICTED r-STIRLING NUMBERS AND THEIR COMBINATORIAL APPLICATIONS

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