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$1\le k\le n$,在一个$n$-置换中,指数为$k$的元素的数的期望为1.(见这里)
证明:
The Cycle index polynomial of $S_n$ is$$
\sum_{\lambda_1 + 2\lambda_2 + ... + n\lambda_n = n} \frac{x_1^{\lambda_1} ... x_n^{\lambda_n}}{1^{\lambda_1} \lambda_1 !2^{\lambda_2}\lambda_2 ! ... n^{\lambda_n} \lambda_n !}$$
所以$k$-cycle的数为$$\sum_{\lambda_1 + 2\lambda_2 + ... + n\lambda_n = n} \frac{\lambda_k}{1^{\lambda_1} \lambda_1 !2^{\lambda_2}\lambda_2 ! ... n^{\lambda_n} \lambda_n !}$$
令$μ_i=λ_i,1\le i\le n,i\ne k;μ_k=λ_k-1$,得
$$\frac{λ_k}{k^{λ_k}·λ_k!}=\frac1k·\frac1{k^{μ_k}μ_k!}$$
所以$$\sum_{\lambda_1 + 2\lambda_2 + ... + n\lambda_n = n} \frac{\lambda_k}{1^{\lambda_1} \lambda_1 !2^{\lambda_2}\lambda_2 ! ... n^{\lambda_n} \lambda_n !}=\frac1k\sum_{\mu_1 + 2\mu_2 + ... + n\mu_n = n} \frac1{1^{\mu_1} \mu_1 !2^{\mu_2}\mu_2 ! ... n^{\mu_n} \mu_n !}=\frac1k$$
所以$k$-cycle的数为$\frac1k$,所以指数为$k$的元素的数的期望为1. |
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