Restricted $r$-Stirling Numbers and their Combinatorial Applications

TL;DR

This paper introduces and studies the generalized $(S,r)$-Stirling numbers, extending classical concepts, and explores their combinatorial applications, including connections to posets, graph enumeration, and generalized poly-Bernoulli numbers.

Contribution

It defines the broad class of $(S,r)$-Stirling numbers and related sequences, providing recurrence relations, matrix inverses, and demonstrating their relevance in combinatorial enumeration.

Findings

$(S,r)$-Stirling numbers generalize classical Stirling numbers.

The inverse matrices encode M"obius functions of posets.

Applications include enumeration of graph structures and generalized number sequences.

Abstract

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 M\"obius 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.