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arXiv · 1710.05103

Exact and asymptotic enumeration of cyclic permutations according to descent set

Sergi Elizalde·Justin M. Troyka
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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 $I \subseteq [n-1]$, 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 the cycles that do not have two consecutive descents.

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Sergi Elizalde, Justin M. Troyka. 2017-10-13. Exact and asymptotic enumeration of cyclic permutations according to descent set. https://doi.org/10.1016/j.jcta.2019.02.012

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