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Generalized Legendre-Stirling Numbers

DOI: 10.4236/ojdm.2014.44014, PP. 109-114

Keywords: Stirling Numbers, Legendre-Stirling Numbers, Set Partitions

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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.


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