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 $\ell_1 + \ell_2 + \cdots + \ell_{\omega(\sigma)} = n$. For an integer $k$ at most $n$, we denote by $s(n,k)$ the number of permutations of $\mathfrak{S}_n$ such that $\omega(\sigma) = k$. Specify $s(n,n)$ and $s(n,1)$ then show that, for $2 \leqslant k \leqslant n-1$, we have $$s(n,k) = s(n-1,k-1) + (n-1)s(n-1,k).$$ For $\sigma \in \mathfrak{S}_n$, one may distinguish the cases $\sigma(1) = 1$ and $\sigma(1) \neq 1$.
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 $\ell_1 + \ell_2 + \cdots + \ell_{\omega(\sigma)} = n$. For an integer $k$ at most $n$, we denote by $s(n,k)$ the number of permutations of $\mathfrak{S}_n$ such that $\omega(\sigma) = k$.
Specify $s(n,n)$ and $s(n,1)$ then show that, for $2 \leqslant k \leqslant n-1$, we have
$$s(n,k) = s(n-1,k-1) + (n-1)s(n-1,k).$$
For $\sigma \in \mathfrak{S}_n$, one may distinguish the cases $\sigma(1) = 1$ and $\sigma(1) \neq 1$.