The partial r-Bell polynomials

TL;DR

This paper generalizes partial Bell polynomials to include various r-analogues of Stirling, Whitney, and Lah numbers, providing new generating functions and combinatorial interpretations.

Contribution

It introduces a unified polynomial framework encompassing multiple r-analogues, with new generating functions and combinatorial insights.

Findings

r-Stirling, r-Whitney, r-Lah, and r-Whitney-Lah numbers are special cases of generalized partial Bell polynomials.

Derived generating functions for various restrictions of these numbers.

Presented new combinatorial interpretations for r-Whitney and r-Whitney-Lah numbers.

Abstract

In this paper, we show that the r-Stirling numbers of both kinds, the r-Whitney numbers of both kinds, the r-Lah numbers and the r-Whitney-Lah numbers form particular cases of family of polynomials forming a generalization of the partial Bell polynomials. We deduce the generating functions of several restrictions of these numbers. In addition, a new combinatorial interpretations is presented for the r-Whitney numbers and the r-Whitney-Lah numbers.