Cyclic permutations: Degrees and combinatorial types

Abstract

This note will give an enumeration of n-cycles in the symmetric group Sn 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 Sn. 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→∞.