Combinatorial interpretations of particular evaluations of complete and elementary symmetric functions

TL;DR

This paper explores how Jacobi-Stirling and Legendre-Stirling numbers relate to elementary and complete symmetric functions, providing new combinatorial interpretations of these special numbers.

Contribution

It reveals that Jacobi-Stirling and Legendre-Stirling numbers are specializations of symmetric functions and offers novel combinatorial interpretations for them.

Findings

Jacobi-Stirling numbers are specializations of elementary symmetric functions.

Legendre-Stirling numbers are specializations of complete symmetric functions.

New combinatorial interpretations for these Stirling numbers are established.

Abstract

The Jacobi-Stirling numbers and the Legendre-Stirling numbers of the first and second kind were first introduced in [6], [7]. In this paper we note that Jacobi-Stirling numbers and Legendre-Stirling numbers are specializations of elementary and complete symmetric functions. We then study combinatorial interpretations of this specialization and obtain new combinatorial interpretations of the Jacobi-Stirling and Legendre-Stirling numbers.