Exact and asymptotic enumeration of cyclic permutations according to 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 $I\subseteq [n-1]$, 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.

31 pages

References in corpus (2)

• Further applications of the Container Method
• Continued fractions for permutation statistics

Cited by in corpus (3)

• Continued fractions for permutation statistics
• Plethystic formulas for permutation enumeration
• Enumerating two permutation classes by the number of cycles