Higher Order r-Dowling polynomials
Funani Sinethemba, Ndiweni Odilo, Nkonkobe Sithembele
TL;DR
This work introduces higher order non-degenerate and degenerate r-Dowling polynomials by leveraging barred preferential arrangements and unfair distributions, framed through generalized Stirling numbers and Bell polynomials. It develops a cohesive combinatorial theory with explicit recurrences, exponential generating functions, integral representations, and asymptotic results, linking the polynomials to Bell polynomials and providing multiple closed-form expressions. The key contributions include the definitions of D^{lambda,x}_{m,r}(n) and D^{lambda,x}_{m,r}(n;alpha), their generating functions, and comprehensive combinatorial proofs of identities, along with asymptotic expansions. The findings deepen the connections between partition-like structures, Dowling-type polynomials, and generalized Stirling numbers, with potential extensions to associated and probabilistic variants and further Bell-type 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.