The $(l,r)$-Stirling numbers: a combinatorial approach

TL;DR

This paper introduces a new generalization of $r$-Stirling numbers called $(l,r)$-Stirling numbers, exploring their combinatorial properties, symmetric function connections, and a limit representation of the multiple zeta function.

Contribution

It presents the $(l,r)$-Stirling numbers, a novel extension of $r$-Stirling numbers, with combinatorial interpretations and applications to special functions.

Findings

Defined $(l,r)$-Stirling numbers of both kinds.

Established combinatorial interpretations and properties.

Derived a limit representation of the multiple zeta function.

Abstract

This work deals with a new generalization of $r$-Stirling numbers using $l$-tuple of permutations and partitions called $(l,r)$-Stirling numbers of both kinds. We study various properties of these numbers using combinatorial interpretations and symmetric functions. Also, we give a limit representation of the multiple zeta function using $(l,r)$-Stirling of the first kind.