On certain combinatorial expansions of the Legendre-Stirling numbers

TL;DR

This paper introduces combinatorial codes for Legendre-Stirling set partitions, enabling new expansions of these numbers and related Jacobi-Stirling numbers, with applications in spectral theory and combinatorics.

Contribution

It provides the first combinatorial codes for Legendre-Stirling set partitions and derives new expansions and grammatical descriptions for related Stirling numbers.

Findings

Combinatorial codes for Legendre-Stirling set partitions

New combinatorial expansions of Legendre-Stirling numbers

Grammatical descriptions of Jacobi-Stirling numbers

Abstract

The Legendre-Stirling numbers of the second kind were introduced by Everitt et al. in the spectral theory of powers of the Legendre differential expressions. In this paper, we provide a combinatorial code for Legendre-Stirling set partitions. As an application, we obtain combinatorial expansions of the Legendre-Stirling numbers of both kinds. Moreover, we present grammatical descriptions of the Jacobi-Stirling numbers of both kinds.