The symmetric and unimodal expansion of Eulerian polynomials via continued fractions

• DOI: 10.1016/j.ejc.2011.08.005
• Publication Date: 2011-10-17T01:44:36Z
• Authors: Heesung Shin; Jiang Zeng
• Keywords: 0102 computer and information sciences; 01 natural sciences; 004; 510; Theoretical Computer Science; Computational Theory and Mathematics; [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]; FOS: Mathematics; Mathematics - Combinatorics; Geometry and Topology; Combinatorics (math.CO); 0101 mathematics

19 pages, 2 figures<br/>This paper was motivated by a conjecture of Br��nd��n (European J. Combin. \textbf{29} (2008), no.~2, 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 (European J. Combin. \textbf{31} (2010), no.~7, 1689--1705.) and that of Josuat-Verg��s (European J. Combin. \textbf{31} (2010), no.~7, 1892--1906).<br/>