Higher Order r-Dowling polynomials

Abstract

Given an ordered set partition, when one insert a number of bars in-between the blocks of the ordered set partition the result is a barred preferential arrangement. In this study, using the notion of barred preferential arrangements we propose a combinatorial interpretation of a type of generalized Bell polynomials. We also define a new higher order generalization of the $r$-Dowling polynomials. We discuss its degenerate and non degenerate versions. Using the notion of barred preferential arrangements we provide a combinatorial interpretation of these higher order $r$-Dowling polynomials. Furthermore, we prove several combinatorial identities on these polynomials. We also provide some integral representations of these polynomials, and provide some of their asymptotic results. We also show several closed form expressions demonstrating how these higher order $r$-Dowling polynomials may be expressed in terms of Bell polynomials.