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r-Lah-Bell numbers

Bn,rLB_{n, r}^L 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.

Formula

BnL=∑k=0n⌊nk⌋rB_n^L = \sum_{k=0}^n\left\lfloor {n \atop k} \right\rfloor_rBnL=∑k=0nn!k!(n+2r−1k+2r−1)B_n^L = \sum_{k=0}^n\frac{n!}{k!} \binom{n + 2r - 1}{k + 2r - 1}

Generating Function

∑n=0∞Bn,rLtnn!=exp(t1−t)∗(11−t)2r\sum_{n=0}^\infty B_{n, r}^L \frac{t^n}{n!} = exp\left(\frac{t}{1-t}\right) * \left(\frac{1}{1-t}\right)^{2r}

References

Kim, Kim, Jang, Lee, Kim: Complete and incomplete Bell polynomials associated with Lah–Bell numbers and polynomials

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