Cyclic Permutations: Degrees and Combinatorial Types

TL;DR

This paper enumerates n-cycles in the symmetric group based on their cyclic descent number, explores their conjugacy classes under rotation, and analyzes the statistical distribution of their degrees, revealing asymptotic normality.

Contribution

It introduces a novel enumeration method for n-cycles by degree, connects these cycles to dynamical systems, and provides statistical insights into their degree distribution.

Findings

Enumeration of n-cycles by cyclic descent number

Relation of cycles to dynamical system orbits

Asymptotic normality of degree distribution

Abstract

This note will give an enumeration of $n$-cycles in the symmetric group ${\mathcal S}_n$ by their degree (also known as their cyclic descent number) and studies similar counting problems for the conjugacy classes of $n$-cycles under the action of the rotation subgroup of ${\mathcal S}_n$. This is achieved by relating such cycles to periodic orbits of an associated dynamical system acting on the circle. We also compute the mean and variance of the degree of a random $n$-cycle and show that its distribution is asymptotically normal as $n \to \infty$.