For any $n\in\mathbb{N}$ let $S\_n$ denote the set of all permutations (bijective maps) $\pi:\{1,\ldots, n\} \to \{1,\ldots,n\}$. For $\pi \in S\_n$ we set $$\text{fix}(\pi) = \{x\in \{1,\ldots, n\}: \pi(x) = x\}.$$


For any $n\in \mathbb{N}$ the expected value of the number of fixed points of a randomly chosen element of $S\_n$ is $$E\_n := \frac{1}{n!}\sum\_{\pi\in\S\_n} |\text{fix}(\pi)|.$$


Is the real sequence $(E\_n)\_{n\in\mathbb{N}}$ bounded? If not, what are the values of $\lim\_{n\to\infty}\frac{E\_n}{n}$ and $\lim\_{n\to\infty}\frac{E\_n}{\log n}$?



Asymptotic behaviour of fixed points in permutations
$E\_n=1$, it is well known and easy to prove by double counting: for any specific point $x$, it is fixed for $(n-1)!$ permutations, now sum up by $x$.


