Restricted $r$-Stirling Numbers and their Combinatorial Applications
Abstract
We study set partitions with distinguished elements and block sizes found in an arbitrary index set . The enumeration of these -partitions leads to the introduction of -Stirling numbers, an extremely wide-ranging generalization of the classical Stirling numbers and the -Stirling numbers. We also introduce the associated -Bell and -factorial numbers. We study fundamental aspects of these numbers, including recurrence relations and determinantal expressions. For with some extra structure, we show that the inverse of the -Stirling matrix encodes the M\"obius functions of two families of posets. Through several examples, we demonstrate that for some the matrices and their inverses involve the enumeration sequences of several combinatorial objects. Further, we highlight how the -Stirling numbers naturally arise in the enumeration of cliques and acyclic orientations of special graphs, underlining their ubiquity and importance. Finally, we introduce related generalizations of the poly-Bernoulli and poly-Cauchy numbers, uniting many past works on generalized combinatorial sequences.
Cite
@article{arxiv.1811.12897,
title = {Restricted $r$-Stirling Numbers and their Combinatorial Applications},
author = {Beáta Bényi and Miguel Méndez and José L. Ramírez and Tanay Wakhare},
journal= {arXiv preprint arXiv:1811.12897},
year = {2018}
}