Exact and asymptotic enumeration of cyclic permutations according to descent set

TL;DR

This paper derives a simple formula for counting cyclic permutations with a specific descent set, and analyzes their asymptotic behavior, including the proportion that are n-cycles and those without consecutive descents.

Contribution

It introduces a new formula linking cyclic permutation counts to ordinary descent numbers and explores their asymptotic properties.

Findings

Almost all permutations with a given descent set are n-cycles asymptotically.

The fraction of permutations that are n-cycles approaches 1/n.

The formula recovers Stanley's result for alternating cycles.

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.