A $q,r$-analogue of poly-Stirling numbers of second kind with combinatorial applications

Takao Komatsu, Eli Bagno, David Garber

Published 2022-09-14Version 1

This paper deals with several generalizations of Stirling number of the second kind, in both analytical and combinatorial directions. Moreover, we present some analytical results regarding generalizations of the Stirling number of the first kind as well. In the analytical part, we generalize the Comtet and Lancaster theorems, which present conditions that are equivalent to the definition of ordinary Stirling numbers of both kinds, to the case of the $q,r$-poly Stirling numbers (which are $q$-analogues of the restricted Stirling numbers defined by Broder and having a polynomial values appearing in their defining recursion). In the combinatorial part, we generalize the approach of Cai-Readdy using restricted growth words in order to represent Stirling numbers of the second kind of Coxeter type $B$, and define a new parameter on restricted growth words of type $B$ that enables us to combinatorially realize some of the identities proven in the analytical part.