Séminaire Lotharingien de Combinatoire, 80B.8 (2018), 12 pp.

Sergi Elizalde and Justin M. Troyka

The Number of Cycles with a Given Descent Set

Abstract. Using a result of Gessel and Reutenauer, we find a simple formula for the number of cyclic permutations with a given descent set, by expressing it in terms of ordinary descent numbers (i.e., those counting all permutations with a given descent set). We then use this formula to show that, for almost all sets I contained in [n-1], the fraction of size-n permutations with descent set I which are n-cycles is asymptotically 1/n. As a special case, we recover a result of Stanley for alternating cycles. We also use our formula to count n-cycles with no two consecutive descents.


Received: November 14, 2017. Accepted: February 17, 2018. Final version: April 1, 2018.

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