Plethystic formulas for permutation enumeration

TL;DR

This paper develops general plethystic formulas for permutation statistics over sets of permutations with symmetric quasisymmetric generating functions, unifying and extending many known enumeration results.

Contribution

It introduces plethystic substitution techniques to derive comprehensive formulas for permutation statistics across various permutation classes.

Findings

Derived formulas for permutation statistics involving descent, peak, and up-down runs.

Unified several known permutation enumeration formulas as special cases.

Applied results to cyclic permutations, involutions, and derangements.

Abstract

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.