Restricted $r$-Stirling numbers of the second kind

In this paper, we introduce the $(r,d)$-Stirling numbers of the second kind, a new generalization of the $r$-Stirling numbers obtained by imposing a distance restriction: for any two elements $i$ and $j$ within the same subset, we require that $|i-j|>d$.

We establish recurrence relations, an explicit formula, ordinary and exponential generating functions, a reduction identity connecting these numbers to the standard $r$-Stirling numbers, and several combinatorial identities.