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Eulerian polynomials and excedance statistics. (English) Zbl 1456.05002

Summary: A formula of Stembridge states that the permutation peak polynomials and descent polynomials are connected via a quadratique transformation. The aim of this paper is to establish the cycle analogue of Stembridge’s formula by using cycle peaks and excedances of permutations. We prove a series of new general formulae expressing polynomials counting permutations by various excedance statistics in terms of refined Eulerian polynomials. Our formulae are comparable with Zhuang’s generalizations [Y. Zhuang, Adv. Appl. Math. 90, 86–144 (2017; Zbl 1366.05010)] 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 [H. Shin and J. Zeng, Eur. J. Comb. 33, No. 2, 111–127 (2012; Zbl 1235.05008)] 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 \(\gamma\)-coefficients of the inversion polynomials restricted on 321-avoiding permutations.

MSC:

05A05 Permutations, words, matrices
05A15 Exact enumeration problems, generating functions
05A19 Combinatorial identities, bijective combinatorics
11B68 Bernoulli and Euler numbers and polynomials
11A55 Continued fractions
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