Eulerian polynomials and excedance statistics

Abstract

A formula of Stembridge states that the permutation peak polynomials and descent polynomials are connected via a quadratique transformation. Rephrasing the latter formula with permutation cycle peaks and excedances we are able to prove a series of general formulas expressing polynomials counting permutations by various excedance statistics in terms of refined Eulerian polynomials. Our formulas are comparable with Zhuang's generalizations [Adv. in Appl. Math. 90 (2017) 86-144] using descent statistics of permutations. Our methods include permutation enumeration techniques involving variations of classical bijections from permutations to Laguerre histories, explicit continued fraction expansions of combinatorial generating functions in Shin and Zeng [European J. Combin. 33 (2012), no. 2, 111-127] and cycle version of modified Foata-Strehl action. We also prove similar formulae for restricted permutations such as derangements and permutations avoiding certain patterns. Moreover, we provide new combinatorial interpretations for the γ-coefficients of the inversion polynomials on 321-avoiding permutations.

A unified approach to combinatorial triangles: a generalized Eulerian polynomial

“…In addition, different generalizations of Eulerian polynomials and Eulerian numbers were considered, see Haglund-Zhang [24], Han-Mao-Zeng [25], Rzadkowski-Urlińska [35], Zhu [48], Zhuang [52]. But recently in many new combinatorial enumerations, there bring out more and more combinatorial triangles satisfying certain three-term recurrence similar to recurrence relations (1.8), (1.9) and (1.11).…”