Let $n$ be a non-zero natural integer. For any permutation $\sigma \in \mathfrak{S}_{n}$, we recall that there exists, up to order, a unique decomposition $\sigma = c_{1} c_{2} \cdots c_{\omega(\sigma)}$, where $\omega(\sigma) \in \mathbb{N}^{*}$ where $c_{1}, \ldots, c_{\omega(\sigma)}$ are cycles with disjoint supports of respective lengths $\ell_{1} \leqslant \ell_{2} \leqslant \cdots \leqslant \ell_{\omega(\sigma)}$ and $\ell_{1} + \ell_{2} + \cdots + \ell_{\omega(\sigma)} = n$. Show that $$\frac{1}{n!} \sum_{\sigma \in \mathfrak{S}_{n}} \omega(\sigma)^{2} \underset{n \rightarrow +\infty}{=} (2\gamma+1)\ln(n) + c + \ln(n)^{2} + O\left(\frac{\ln(n)}{n}\right)$$ for a real number $c$ to be specified.
Let $n$ be a non-zero natural integer. For any permutation $\sigma \in \mathfrak{S}_{n}$, we recall that there exists, up to order, a unique decomposition $\sigma = c_{1} c_{2} \cdots c_{\omega(\sigma)}$, where $\omega(\sigma) \in \mathbb{N}^{*}$ where $c_{1}, \ldots, c_{\omega(\sigma)}$ are cycles with disjoint supports of respective lengths $\ell_{1} \leqslant \ell_{2} \leqslant \cdots \leqslant \ell_{\omega(\sigma)}$ and $\ell_{1} + \ell_{2} + \cdots + \ell_{\omega(\sigma)} = n$.
Show that
$$\frac{1}{n!} \sum_{\sigma \in \mathfrak{S}_{n}} \omega(\sigma)^{2} \underset{n \rightarrow +\infty}{=} (2\gamma+1)\ln(n) + c + \ln(n)^{2} + O\left(\frac{\ln(n)}{n}\right)$$
for a real number $c$ to be specified.