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 $n\to \infty$.

中文翻译：

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循环排列：度数和组合类型

本笔记将列举对称群中的n 个循环${\mathcal{秒}}_n$根据它们的度数（也称为它们的循环下降数），并研究了n循环的共轭类在${\mathcal{秒}}_n$. 这是通过将这些循环与作用在该圆上的相关动力系统的周期轨道相关联来实现的。我们还计算了随机n循环次数的均值和方差，并表明它的分布是渐近正态的$n\to \infty$.