Generalized Legendre-Stirling Numbers

• 作者: K. C. Garrett Kendra Killpatrick
• 来源期刊: 离散数学期刊(英文)
• 学科: 数学
• 关键词: STIRLING NUMBERS Legendre-Stirling NUMBERS SET PARTITIONS
• 年，卷（期）: 2014,（4）
• 页码范围: 109-114
• 页数: 6页
• 分类号: O1

摘要

The Legendre-Stirling numbers were discovered by Everitt, Littlejohn and Wellman in 2002 in a study of the spectral theory of powers of the classical second-order Legendre differential operator. In 2008, Andrews and Littlejohn gave a combinatorial interpretation of these numbers in terms of set partitions. In 2012, Mongelli noticed that both the Jacobi-Stirling and the Legendre-Stirling numbers are in fact specializations of certain elementary and complete symmetric functions and used this observation to give a combinatorial interpretation for the generalized Legendre-Stirling numbers. In this paper we provide a second combinatorial interpretation for the generalized Legendre-Stirling numbers which more directly generalizes the definition of Andrews and Littlejohn and give a combinatorial bijection between our interpretation and the Mongelli interpretation. We then utilize our interpretation to prove a number of new identities for the generalized Legendre-Stirling numbers.