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(S, r)-Lah-Bell numbers

LS,r(n)L_{S, r}(n) is the number of partitions of an (n+r)(n + r) element set into lists, where a list means a non-empty, linearly ordered subset, such that rr distinguished elements have to be in distinct ordered blocks, and each block has cardinality belonging to some set SS.

Generating Function

∑n=0∞LS,r(n)xnn!=exp⁡(∑s∈Sxs)(∑s∈Ssxs−1)r\sum_{n=0}^\infty L_{S, r}(n) \frac{x^n}{n!} = \exp\left( \sum_{s \in S} x^s \right) \left( \sum_{s \in S} s x^{s-1} \right)^r

References

Bényi, Méndez, Ramirez: GENERALIZED ORDERED SET PARTITIONS

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