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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