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The $ q$-tangent and $ q$-secant numbers via basic Eulerian polynomials


Authors: Dominique Foata and Guo-Niu Han
Journal: Proc. Amer. Math. Soc. 138 (2010), 385-393
MSC (2000): Primary 05A15, 05A30, 05E15
DOI: https://doi.org/10.1090/S0002-9939-09-10144-2
Published electronically: October 2, 2009
MathSciNet review: 2557155
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Abstract: The classical identity that relates Eulerian polynomials to tangent numbers together with the parallel result dealing with secant numbers is given a $ q$-extension, both analytically and combinatorially. The analytic proof is based on a recent result by Shareshian and Wachs and the combinatorial one on the geometry of alternating permutations.


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Additional Information

Dominique Foata
Affiliation: Institut Lothaire, 1 rue Murner, F-67000 Strasbourg, France
Email: foata@math.u-strasbg.fr

Guo-Niu Han
Affiliation: Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René-Descartes, F-67084 Strasbourg, France
Email: guoniu@math.u-strasbg.fr

DOI: https://doi.org/10.1090/S0002-9939-09-10144-2
Keywords: $q$-tangent numbers, $q$-secant numbers, $q$-Eulerian polynomials, excedances, derangements, desarrangements, alternating permutations.
Received by editor(s): October 6, 2008
Published electronically: October 2, 2009
Communicated by: Jim Haglund
Article copyright: © Copyright 2009 American Mathematical Society

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