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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