Justify that $\mathcal{X}_n$ is a finite set and determine its cardinality.

What is the cardinality of $\mathcal{M}_{n}(\{-1,1\})$? Is this set a vector subspace of $\mathcal{M}_{n}(\mathbb{R})$?

Calculate $$\sum_{\sigma \in \mathfrak{S}_{n}} \varepsilon(\sigma), \quad \sum_{\sigma \in \mathfrak{S}_{n}} \varepsilon(\sigma) \nu(\sigma) \quad \text{and} \quad \sum_{\sigma \in \mathfrak{S}_{n}} \frac{\varepsilon(\sigma)}{\nu(\sigma)+1}.$$

Establish that $$\operatorname{Card}\left\{\sigma \in \mathfrak{S}_{n} : \varepsilon(\sigma) = 1\right\} = \operatorname{Card}\left\{\sigma \in \mathfrak{S}_{n} : \varepsilon(\sigma) = -1\right\}$$ and deduce the probability that a permutation of $\mathfrak{S}_{n}$ drawn uniformly at random has a prescribed signature.

For $\sigma \in \mathfrak{S}_{n}$, specify the condition on $\nu(\sigma)$ for which $\sigma \in \mathfrak{D}_{n}$. Deduce that $$\operatorname{Card}\left\{\sigma \in \mathfrak{D}_{n} : \varepsilon(\sigma) = 1\right\} = \operatorname{Card}\left\{\sigma \in \mathfrak{D}_{n} : \varepsilon(\sigma) = -1\right\} + (-1)^{n-1}(n-1).$$

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 $\omega(\sigma) \in \mathbb{N}^{*}$ 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 $\ell_{1} + \ell_{2} + \cdots + \ell_{\omega(\sigma)} = n$. 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 $\sigma(1) \neq 1$.

Calculate $$\sum_{\sigma \in \mathfrak{S}_n} \varepsilon(\sigma), \quad \sum_{\sigma \in \mathfrak{S}_n} \varepsilon(\sigma) \nu(\sigma) \quad \text{and} \quad \sum_{\sigma \in \mathfrak{S}_n} \frac{\varepsilon(\sigma)}{\nu(\sigma)+1}.$$

Establish that $$\operatorname{Card}\left\{\sigma \in \mathfrak{S}_n : \varepsilon(\sigma) = 1\right\} = \operatorname{Card}\left\{\sigma \in \mathfrak{S}_n : \varepsilon(\sigma) = -1\right\}$$ and deduce the probability that a permutation of $\mathfrak{S}_n$ drawn uniformly at random has a prescribed signature.

For $\sigma \in \mathfrak{S}_n$, specify the condition on $\nu(\sigma)$ for which $\sigma \in \mathfrak{D}_n$. Deduce that $$\operatorname{Card}\left\{\sigma \in \mathfrak{D}_n : \varepsilon(\sigma) = 1\right\} = \operatorname{Card}\left\{\sigma \in \mathfrak{D}_n : \varepsilon(\sigma) = -1\right\} + (-1)^{n-1}(n-1).$$

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 $\sigma(1) \neq 1$.

Let $n$ be a non-zero natural integer. For an integer $k$ at most $n$, we denote by $s(n,k)$ the number of permutations of $\mathfrak{S}_n$ such that $\omega(\sigma) = k$.
Establish that, for any real $x$, $$\prod_{i=0}^{n-1}(x+i) = \sum_{k=1}^{n} s(n,k) x^k.$$