论文标题
通过给定循环类型的随机排列产生的概率
Probability of generation by random permutations of given cycle type
论文作者
论文摘要
假设$π$和$π'$是$ s_n $的两个随机元素，具有约束周期类型，使得$π$具有$ x n^{1/2} $固定点和$ yn/2 $ the/2 $ twip-twip-cycles，并且同样有$ x'$具有$ x'n^{1/2} $ y'n n n^points和$ y'n n/2 $两cycles。我们表明，$ g = \langleπ，π'\ rangle $是传递的事件，$ g \ geq a_n $都具有大约\ [（1- yy'）^{1/2} {1/2} {1/2} {1/2} \ exp \ left（ - \ \ \ \ \ frac {xx' + \ frac12 x^2 x^2 y' + \ frac12 Y' yy'} \ right），\]提供$（x，x'）$不接近$（0，\ infty）$或$（\ infty，0）$。该公式来自作者最近的一篇论文（Arxiv：1904.12180）的一些初步结果。作为一个应用程序，我们表明，$ s_n $的两个均匀随机结合类别的两个均匀随机元素以约51％的概率生成该组。
Suppose $π$ and $π'$ are two random elements of $S_n$ with constrained cycle types such that $π$ has $x n^{1/2}$ fixed points and $yn/2$ two-cycles, and likewise $π'$ has $x' n^{1/2}$ fixed points and $y'n/2$ two-cycles. We show that the events that $G = \langle π, π' \rangle$ is transitive and $G \geq A_n$ both have probability approximately \[(1 - yy')^{1/2} \exp\left(- \frac{xx' + \frac12 x^2 y' + \frac12 {x'}^2 y}{1 - yy'}\right),\] provided $(x, x')$ is not close to $(0, \infty)$ or $(\infty, 0)$. This formula is derived from some preliminary results in a recent paper (arXiv:1904.12180) of the authors. As an application, we show that two uniformly random elements of uniformly random conjugacy classes of $S_n$ generate the group with probability about 51%.
