r-Lah numbers

$$\left\lfloor {n \atop k} \right\rfloor_r$$ is the number of partitions of an $n$ element set into $k$lists, where a list means a non-empty, linearly ordered subset such that $r$ distinguished elements are in distinct ordered blocks.

Cases

• Case $r=2$: OEIS A143497
• Case $r=3$: OEIS A143498
• Case $r=4$: OEIS A143499

Recurrence

• $$\left\lfloor {n+1 \atop k} \right\rfloor_r = \left\lfloor {n \atop k-1} \right\rfloor_r + (n + k + 2r) \left\lfloor {n \atop k} \right\rfloor_r \text{ for } 1 \leq k \leq n \text{ and } r \geq 0$$

Formulas

• $$\left\lfloor {n \atop k} \right\rfloor_r = \frac{n!}{k!} \binom{n + 2r - 1}{k + 2r - 1}$$

Generating Function

$$\sum_{n=k}^\infty \left\lfloor {n \atop k} \right\rfloor_r \frac{x^n}{n!} = \frac{1}{k!} \left( \frac{x}{1-x} \right)^k \left( \frac{1}{1-x} \right)^{2r}$$

Source

Nyul, Rácz: The r-Lah numbers