A combinatorial interpretation of the Legendre-Stirling numbers

Abstract

Abstract. The Legendre-Stirling numbers were discovered in 2002 as a result of a problem involving the spectral theory of powers of the classical secondorder Legendre differential expression. Specifically, these numbers are the coefficients of integral composite powers of the Legendre expression in Lagrangian symmetric form. Quite remarkably, they share many similar properties with the classical Stirling numbers of the second kind which, as shown by Littlejohn and Wellman, are the coefficients of integral powers of the Laguerre differential expression. An open question regarding the Legendre-Stirling numbers has been to obtain a combinatorial interpretation of these numbers. In this paper, we provide such an interpretation.

The Legendre–Stirling numbers

“…For an excellent account of Stirling numbers of the first and second kind, see Comtet's text [5,Chapter V]. This paper is a continuation of the recent work of Andrews and Littlejohn in [3], where the authors obtained a combinatorial interpretation of the Legendre-Stirling numbers. The contents of this paper are as follows.…”

“…It is natural to ask: what do the Legendre-Stirling numbers count? Andrews and Littlejohn gave a combinatorial interpretation of these numbers in [3].…”

“…Section 3 deals with an alternative formula for the Legendre-Stirling numbers; as we will see, this formula has its advantages over the one given in (1.3). In Section 4, we give the combinatorial interpretation of the Legendre-Stirling numbers that is developed in [3]. Section 5 deals with specific combinatorial properties of the Legendre-Stirling numbers and how these properties compare with those of the Stirling numbers of the second kind.…”

γ-positivity and partial γ-positivity of descent-type polynomials

“…and satisfy the recurrence relation LS (n, k) = LS (n − 1, k − 1) + k(k + 1)LS (n − 1, k), with the initial conditions LS (0, 0) = 1 and LS (0, k) = 0 for k ≥ 1. Andrews and Littlejohn [3] discovered that LS (n, k) is the number of Legendre-Stirling set partitions of the set {1 1 , 1 2 , 2 1 , 2 2 , . .…”

On Certain Combinatorial Expansions of the Legendre-Stirling Numbers

“…In particular, let <> denote the empty zero box. For example, {1, 1, 3}{2, 2} < 3 >∈ LS (3,2). A classical result of Andrews and Littlejohn [3, Theorem 2] says that LS (n, k) = #LS(n, k).…”