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 $\omega(\sigma) \in \mathbb{N}^*$ and $c_1, \ldots, c_{\omega(\sigma)}$ are cycles with disjoint supports of respective lengths $\ell_1 \leqslant \ell_2 \leqslant \cdots \leqslant \ell_{\omega(\sigma)}$ and $\ell_1 + \ell_2 + \cdots + \ell_{\omega(\sigma)} = n$. We thus obtain a map $\omega : \mathfrak{S}_n \rightarrow \mathbb{N}$. For an integer $k$ at most $n$, we denote by $s(n,k)$ the number of permutations of $\mathfrak{S}_n$ such that $\omega(\sigma) = k$. We consider, on the probability space $(\mathfrak{S}_n, \mathscr{P}(\mathfrak{S}_n))$ equipped with the uniform probability, the random variable $X_n$ defined by $X_n(\sigma) = \omega(\sigma)$. Calculate, for $n \in \{2, 3, 4\}$, the quantity $\frac{1}{n!} \sum_{\sigma \in \mathfrak{S}_n} \omega(\sigma)$.
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 $\omega(\sigma) \in \mathbb{N}^*$ and $c_1, \ldots, c_{\omega(\sigma)}$ are cycles with disjoint supports of respective lengths $\ell_1 \leqslant \ell_2 \leqslant \cdots \leqslant \ell_{\omega(\sigma)}$ and $\ell_1 + \ell_2 + \cdots + \ell_{\omega(\sigma)} = n$. We thus obtain a map $\omega : \mathfrak{S}_n \rightarrow \mathbb{N}$. For an integer $k$ at most $n$, we denote by $s(n,k)$ the number of permutations of $\mathfrak{S}_n$ such that $\omega(\sigma) = k$. We consider, on the probability space $(\mathfrak{S}_n, \mathscr{P}(\mathfrak{S}_n))$ equipped with the uniform probability, the random variable $X_n$ defined by $X_n(\sigma) = \omega(\sigma)$.
Calculate, for $n \in \{2, 3, 4\}$, the quantity $\frac{1}{n!} \sum_{\sigma \in \mathfrak{S}_n} \omega(\sigma)$.