Let $n \in \mathbb{N}^*$. For $k \in \mathbb{N}$, we denote $\mathcal{B}(n, k)$ the set of maps $\sigma \in \operatorname{MD}(n+1)$ such that $\sigma(2) - \sigma(1) = k+1$. For $k \in \mathbb{N}$ and $s \in \mathbb{N}$, we denote $\mathcal{C}(n, s, k)$ the set of elements $\sigma$ of $\operatorname{MD}(n+2)$ such that $\sigma(2) - \sigma(1) = s+1, \quad n+2-\sigma(2) = k$. We denote by $\Psi_{n,s,k}$ the map from $\mathcal{C}(n,s,k)$ to $\mathcal{B}(n,k)$ defined by $\Psi_{n,s,k}(\sigma) = \eta$. 4. Let $\eta \in \mathcal{B}(n, k)$ and let $\xi = \operatorname{Opp}(\eta)$. a. Verify that the number $m$ of integers $j \geq 3$ such that $\xi(j) > \xi(2)$ satisfies $m \geq k$. We denote these integers by $j_1, \ldots, j_m$, with $\xi(j_1) > \xi(j_2) \cdots > \xi(j_m)$. b. We set $u_2 = \xi(j_k) - \frac{1}{2} > \xi(2)$. Show that the number $m'$ of integers $i \geq 2$ such that $\xi(i) < u_2$ satisfies $m' \geq s$. We denote them by $i_1, \ldots, i_{m'}$, with $\xi(i_1) > \cdots > \xi(i_{m'})$ and we set $u_1 = \xi(i_k) - \frac{1}{2}$. c. By considering the map $\theta$ defined by $$\theta(1) = u_1, \theta(2) = u_2, \theta(3) = \xi(2), \ldots, \theta(n+2) = \xi(n+1)$$ show the existence of $\sigma \in \mathcal{C}(n, s, k)$ satisfying $\Psi_{n,s,k}(\sigma) = \eta$. d. Show that $\Psi_{n,s,k}$ is bijective.
Let $n \in \mathbb{N}^*$. For $k \in \mathbb{N}$, we denote $\mathcal{B}(n, k)$ the set of maps $\sigma \in \operatorname{MD}(n+1)$ such that $\sigma(2) - \sigma(1) = k+1$. For $k \in \mathbb{N}$ and $s \in \mathbb{N}$, we denote $\mathcal{C}(n, s, k)$ the set of elements $\sigma$ of $\operatorname{MD}(n+2)$ such that $\sigma(2) - \sigma(1) = s+1, \quad n+2-\sigma(2) = k$.
We denote by $\Psi_{n,s,k}$ the map from $\mathcal{C}(n,s,k)$ to $\mathcal{B}(n,k)$ defined by $\Psi_{n,s,k}(\sigma) = \eta$.
4. Let $\eta \in \mathcal{B}(n, k)$ and let $\xi = \operatorname{Opp}(\eta)$.\\
a. Verify that the number $m$ of integers $j \geq 3$ such that $\xi(j) > \xi(2)$ satisfies $m \geq k$. We denote these integers by $j_1, \ldots, j_m$, with $\xi(j_1) > \xi(j_2) \cdots > \xi(j_m)$.\\
b. We set $u_2 = \xi(j_k) - \frac{1}{2} > \xi(2)$. Show that the number $m'$ of integers $i \geq 2$ such that $\xi(i) < u_2$ satisfies $m' \geq s$. We denote them by $i_1, \ldots, i_{m'}$, with $\xi(i_1) > \cdots > \xi(i_{m'})$ and we set $u_1 = \xi(i_k) - \frac{1}{2}$.\\
c. By considering the map $\theta$ defined by
$$\theta(1) = u_1, \theta(2) = u_2, \theta(3) = \xi(2), \ldots, \theta(n+2) = \xi(n+1)$$
show the existence of $\sigma \in \mathcal{C}(n, s, k)$ satisfying $\Psi_{n,s,k}(\sigma) = \eta$.\\
d. Show that $\Psi_{n,s,k}$ is bijective.