Eulerian polynomials and excedance statistics

TL;DR

This paper establishes new formulas connecting cycle peaks and excedances of permutations with Eulerian polynomials, extending Stembridge's results and providing combinatorial interpretations for related coefficients.

Contribution

It introduces cycle analogues of Stembridge's formula using cycle peaks and excedances, and develops new combinatorial and algebraic tools for permutation enumeration.

Findings

Derived formulas linking permutation statistics to Eulerian polynomials.

Extended results to restricted permutations like derangements and pattern-avoiding permutations.

Provided combinatorial interpretations for gamma-coefficients in specific polynomial classes.

Abstract

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 [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 $\gamma$-coefficients of the inversion polynomials restricted on $321$-avoiding permutations.