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Séminaire Lotharingien de Combinatoire, 84B.12 (2020), 12 pp.

# Ira M. Gessel and Yan Zhuang

# Counting Permutations by Peaks, Descents, and Cycle Type

**Abstract.**
We derive a general formula describing the joint distribution of two permutation statistics - the peak number
and the descent number - over any set of permutations whose quasisymmetric generating function is a symmetric function.
Our formula involves a certain kind of plethystic substitution on quasisymmetric generating functions. We apply this result
to cyclic permutations, involutions, and derangements, and to give a generating function formula for counting permutations
by peaks, descents, and cycle type. We recover as special cases results previously derived by Désarménien-Foata,
Gessel-Reutenauer, Fulman, and Diaconis-Fulman-Holmes.

Received: November 20, 2019.
Accepted: February 20, 2020.
Final version: April 30, 2020.

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