循环置换：度与组合类型
Cyclic Permutations: Degrees and Combinatorial Types
AI总结：
本文枚举对称群中n-循环按度的计数，通过动力系统周期轨道研究共轭类计数，并证明随机n-循环的度分布渐近正态。
摘要：
本文给出对称群 ${\mathcal S}_n$ 中 $n$-循环按度（也称为循环下降数）的计数，并研究在 ${\mathcal S}_n$ 的旋转子群作用下 $n$-循环共轭类的类似计数问题。这是通过将此类循环与作用在圆上的相关动力系统的周期轨道联系起来实现的。我们还计算了随机 $n$-循环的度的均值和方差，并证明当 $n \to \infty$ 时其分布渐近正态。
英文摘要：
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 $n \to \infty$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
