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 $\sigma(2) - \sigma(1) = s+1, \quad n+2-\sigma(2) = k$. 3. In this question and the next, we fix $n \geq 2$, $1 \leq k \leq n-1$ and $1 \leq s \leq n-k$. We propose to construct a bijection from $\mathcal{C}(n, s, k)$ to $\mathcal{B}(n, k)$. Let $\sigma \in \mathcal{C}(n, s, k)$. a. Verify that the number $m$ of integers $j \geq 4$ such that $\sigma(j) > \sigma(3)$ satisfies $m \geq k$. We denote $j_1, \ldots, j_m$ these integers, which we order in such a way that $\sigma(j_1) < \sigma(j_2) < \cdots < \sigma(j_m)$. b. Show that $\xi$ defined by $\xi(1) = \sigma(j_k) + \frac{1}{2}$, $\xi(2) = \sigma(3)$, $\xi(n+1) = \ldots$ satisfies $$\xi(p) > \xi(p+1) \text{ for } p \text{ odd}, \quad \xi(p) < \xi(p+1) \text{ for } p \text{ even}$$ and that the interval $]\xi(2), \xi(1)[$ contains exactly $k$ elements of $\{\xi(3), \ldots, \xi(n+1)\}$. c. We denote $A = \xi(\Delta_{n+1})$ and we set $\bar{\xi} = \beta_A \circ \xi$ (we recall that $\beta_A$ denotes the unique increasing bijection from $A$ to $\Delta_{n+1}$). Show that $\bar{\xi} \in \operatorname{DM}(n+1)$. d. Let $\eta = \operatorname{Opp}(\bar{\xi})$. Verify that $\eta \in \mathcal{B}(n, k)$.
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 $\sigma(2) - \sigma(1) = s+1, \quad n+2-\sigma(2) = k$.
3. In this question and the next, we fix $n \geq 2$, $1 \leq k \leq n-1$ and $1 \leq s \leq n-k$. We propose to construct a bijection from $\mathcal{C}(n, s, k)$ to $\mathcal{B}(n, k)$. Let $\sigma \in \mathcal{C}(n, s, k)$.\\
a. Verify that the number $m$ of integers $j \geq 4$ such that $\sigma(j) > \sigma(3)$ satisfies $m \geq k$. We denote $j_1, \ldots, j_m$ these integers, which we order in such a way that $\sigma(j_1) < \sigma(j_2) < \cdots < \sigma(j_m)$.\\
b. Show that $\xi$ defined by $\xi(1) = \sigma(j_k) + \frac{1}{2}$, $\xi(2) = \sigma(3)$, $\xi(n+1) = \ldots$ satisfies
$$\xi(p) > \xi(p+1) \text{ for } p \text{ odd}, \quad \xi(p) < \xi(p+1) \text{ for } p \text{ even}$$
and that the interval $]\xi(2), \xi(1)[$ contains exactly $k$ elements of $\{\xi(3), \ldots, \xi(n+1)\}$.\\
c. We denote $A = \xi(\Delta_{n+1})$ and we set $\bar{\xi} = \beta_A \circ \xi$ (we recall that $\beta_A$ denotes the unique increasing bijection from $A$ to $\Delta_{n+1}$). Show that $\bar{\xi} \in \operatorname{DM}(n+1)$.\\
d. Let $\eta = \operatorname{Opp}(\bar{\xi})$. Verify that $\eta \in \mathcal{B}(n, k)$.