Combinatorial interpretations of the Jacobi-Stirling numbers

TL;DR

This paper provides combinatorial interpretations for the Jacobi-Stirling numbers of both kinds, unifying their theory with classical Stirling and central factorial numbers, thus enriching their combinatorial understanding.

Contribution

It introduces new combinatorial interpretations for Jacobi-Stirling numbers, linking them to classical sequences and offering a unified combinatorial framework.

Findings

Unified combinatorial interpretations for Jacobi-Stirling numbers of both kinds.

Connections established between Jacobi-Stirling numbers, Stirling numbers, and central factorial numbers.

Enhanced understanding of the combinatorial structure of these polynomial refinements.

Abstract

The Jacobi-Stirling numbers of the first and second kinds were introduced in 2006 in the spectral theory and are polynomial refinements of the Legendre-Stirling numbers. Andrews and Littlejohn have recently given a combinatorial interpretation for the second kind of the latter numbers. Noticing that these numbers are very similar to the classical central factorial numbers, we give combinatorial interpretations for the Jacobi-Stirling numbers of both kinds, which provide a unified treatment of the combinatorial theories for the two previous sequences and also for the Stirling numbers of both kinds.