要求

Problem 1

• 计算${\displaystyle T_{n}}$是多少；
• ${\displaystyle n\to \infty }$时，随机重排并发回作业后，满足上述要求的概率是多少。

Problem 2

Let ${\displaystyle \pi }$ be a permutation of ${\displaystyle [n]}$. Recall that a cycle of permutation ${\displaystyle \pi }$ of length ${\displaystyle k}$ is a tuple ${\displaystyle (a_{1},a_{2},\ldots ,a_{k})}$ such that ${\displaystyle a_{2}=\pi (a_{1}),a_{3}=\pi (a_{2}),\ldots ,a_{k}=\pi (a_{k-1})}$ and ${\displaystyle a_{1}=\pi (a_{k})\,}$. Thus a fixed point of ${\displaystyle \pi }$ is just a cycle of length 1.

• Fix ${\displaystyle k\geq 1}$. Let ${\displaystyle f_{k}(n)}$ be the number of permutations of ${\displaystyle [n]}$ having no cycle of length ${\displaystyle k}$. Compute this ${\displaystyle f_{k}(n)}$ and the limit ${\displaystyle \lim _{n\rightarrow \infty }{\frac {f_{k}(n)}{n!}}}$.

Problem 3

Let ${\displaystyle N_{m}}$ denote the number of objects from a collection of ${\displaystyle N}$ objects that possess exactly ${\displaystyle m}$ of the properties ${\displaystyle a_{1},a_{2},\ldots ,a_{r}}$. Generalize the principle of inclusion-exclusion by computing ${\displaystyle N_{m}}$ as the following form

${\displaystyle N_{m}=\sum _{k=m}^{r}(-1)^{k-m}{k \choose m}s_{k}}$,
and please explicitly give the ${\displaystyle s_{k}}$.

Problem 4

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• 定义这个对称构成的群，可以通过生成元定义，也可以直接把元素都写出来；
• 写出cycle index；
• 如果要求三种颜色出现的次数一样多，写出这时的pattern inventory；
• 如果有四种颜色红、绿、蓝、黄，并要求四种颜色出现的次数一样多，写出这时的pattern inventory；