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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A combinatorial interpretation of the Legendre-Stirling numbers
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by George E. Andrews and Lance L. Littlejohn PDF
Proc. Amer. Math. Soc. 137 (2009), 2581-2590 Request permission


The Legendre-Stirling numbers were discovered in 2002 as a result of a problem involving the spectral theory of powers of the classical second-order 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.
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Additional Information
  • George E. Andrews
  • Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16801
  • MR Author ID: 26060
  • Email:
  • Lance L. Littlejohn
  • Affiliation: Department of Mathematics, Baylor University, One Bear Place #97328, Waco, Texas 76798-7328
  • Email:
  • Received by editor(s): September 2, 2008
  • Received by editor(s) in revised form: October 21, 2008
  • Published electronically: February 17, 2009
  • Communicated by: Jim Haglund
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 2581-2590
  • MSC (2000): Primary 05A05, 05A15, 33C45; Secondary 34B24, 34L05, 47E05
  • DOI:
  • MathSciNet review: 2497469