Cyclic Permutations: Degrees and Combinatorial Types

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 $n\to \infty$.

29 pages, 3 figures. Mildly edited with updated section 4 and figure 1. To appear in J. Comb. Theory Ser. A

References in corpus (2)

• A central limit theorem for descents and major indices in fixed conjugacy classes of $S_n$
• On Combinatorial Types of Periodic Orbits of the Map $x\mapsto kx$ (mod $\mathbb{Z}$)