Restricted r-Stirling numbers and their combinatorial applications

We study set partitions with r distinguished elements and block sizes found in an arbitrary index set S. The enumeration of these (S, r)-partitions leads to the introduction of (S, r)-Stirling numbers, an extremely wide-ranging generalization of the classical Stirling numbers and the r-Stirling numbers. We also introduce the associated (S, r)-Bell and (S, r)-factorial numbers. We study fundamental aspects of these numbers, including recurrence relations and determinantal expressions. For S with some extra structure, we show that the inverse of the (S, r)-Stirling matrix encodes the Möbius functions of two families of posets. Through several examples, we demonstrate that for some S the matrices and their inverses involve the enumeration sequences of several combinatorial objects. Further, we highlight how the (S, r)-Stirling numbers naturally arise in the enumeration of cliques and acyclic orientations of special graphs, underlining their ubiquity and importance. Finally, we introduce related (S, r) generalizations of the poly-Bernoulli and poly-Cauchy numbers, uniting many past works on generalized combinatorial sequences.