Higher Order r-Dowling polynomials

TL;DR

This paper introduces a higher order generalization of r-Dowling polynomials, providing combinatorial interpretations, identities, integral representations, asymptotic analysis, and connections to Bell polynomials.

Contribution

It proposes a new higher order r-Dowling polynomial, offering combinatorial, algebraic, and analytical insights, extending existing polynomial families.

Findings

Combinatorial interpretation via barred preferential arrangements

Derivation of identities and integral representations

Asymptotic behavior and closed-form expressions

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.