For $n\in\mathbb{N}$, let $[n]:= \{1,\ldots,n\}$. Let $\text{Fun}(n)$ denote the set of all functions $f:[n]\to[n]$. To $f\in\text{Fun}(n)$ associate a sequence $\text{seq}(f))$ defined recursively by

- $\text{seq}(f)_1 = f(1)$, and
- $\text{seq}(f)_{k+1} = f(\text{seq}(f)_k)$ for all $k\in\mathbb{N}$.

Eventually $\text{seq}(f)$ will be periodic, and with $\text{per}(f)$ we denote the length of the period of $\text{seq}(f)$. By $E_n$ we denote the *expected value* of $\text{per}(f)$ for any $f\in\text{Fun}(n)$. Explicitly, we have $$E_n = \frac{1}{n^n}\sum_{f\in\text{Fun}(n)}\text{per}(f).$$

**Questions.** Do we have $\lim\sup_{n\to\infty} E_n/n > 0$, and if yes, what is that value? If no, do we have $\lim\sup_{n\to\infty} E_n/\log(n) > 0$?