We prove several general formulas for the distributions of various permutation statistics over any set of permutations whose quasisymmetric generating function is a symmetric function. Our formulas involve certain kinds of plethystic substitutions on quasisymmetric generating functions, and the permutation statistics we consider include the descent number, peak number, left peak number, and the number of up-down runs. We apply these results to cyclic permutations, involutions, and derangements, and more generally, to derive formulas for counting all permutations by the above statistics jointly with the number of fixed points and jointly with cycle type. A number of known formulas are recovered as special cases of our results, including formulas of D\'esarm\'enien-Foata, Gessel-Reutenauer, Stembridge, Fulman, Petersen, Diaconis-Fulman-Holmes, Zhuang, and Athanasiadis.

中文翻译：

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置换枚举的丰满公式

我们证明了在准对称生成函数是对称函数的任何排列集合上各种排列统计量分布的几个通用公式。我们的公式涉及准对称生成函数上的某些类型的 plethystic 替换，我们考虑的置换统计包括下降数、峰数、左峰数和上下运行数。我们将这些结果应用于循环排列、对合和乱序，更一般地说，通过上述统计数据以及固定点的数量和循环类型，推导出计算所有排列的公式。许多已知公式被恢复为我们结果的特例，包括 D\'esarm\'enien-Foata、Gessel-Reutenauer、Stembridge、Fulman、Petersen 的公式，