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

中文翻译：

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欧拉多项式和超越统计

Stembridge 的一个公式指出置换峰值多项式和下降多项式通过二次变换连接。用置换循环峰值和超出数重新表述后一个公式，我们能够证明一系列通用公式，这些公式表示通过各种超出数统计排列的多项式，根据精炼的欧拉多项式。我们的公式与庄的概括[Adv. 在应用程序中。数学。90 (2017) 86-144] 使用排列的下降统计。我们的方法包括置换枚举技术，涉及从置换到 Laguerre 历史的经典双射的变化，Shin 和 Zeng 中组合生成函数的显式连分数扩展~[European J. Combin. 33 (2012)，没有。2、111--127] 和修改的 Foata-Strehl 动作的循环版本。我们还证明了限制排列的类似公式，例如避免某些模式的乱序和排列。此外，我们为$321$-避免排列的反演多项式的$\gamma$-系数提供了新的组合解释。