The symmetric and unimodal expansion of Eulerian polynomials via continued fractions |
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Authors: | Heesung Shin |
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Institution: | a Department of Mathematics, Inha University, 253 Yonhyun-dong, Nam-gu, Incheon, 402-751, South Koreab Université de Lyon; Université Lyon 1, Institut Camille Jordan; UMR 5208 du CNRS; 43, boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France |
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Abstract: | This paper was motivated by a conjecture of Brändén P. Brändén, Actions on permutations and unimodality of descent polynomials, European J. Combin. 29 (2) (2008) 514-531] about the divisibility of the coefficients in an expansion of generalized Eulerian polynomials, which implies the symmetric and unimodal property of the Eulerian numbers. We show that such a formula with the conjectured property can be derived from the combinatorial theory of continued fractions. We also discuss an analogous expansion for the corresponding formula for derangements and prove a (p,q)-analogue of the fact that the (-1)-evaluation of the enumerator polynomials of permutations (resp. derangements) by the number of excedances gives rise to tangent numbers (resp. secant numbers). The (p,q)-analogue unifies and generalizes our recent results H. Shin, J. Zeng, The q-tangent and q-secant numbers via continued fractions, European J. Combin. 31 (7) (2010) 1689-1705] and that of Josuat-Vergès M. Josuat-Vergés, A q-enumeration of alternating permutations, European J. Combin. 31 (7) (2010) 1892-1906]. |
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