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 $\sigma(2) - \sigma(1) = s+1, \quad n+2-\sigma(2) = k$.
1. For $m \geq 2$ verify that the map $\operatorname{Opp} : \Sigma_m \rightarrow \Sigma_m$, which to $\sigma \in \Sigma_m$ associates $\eta \in \Sigma_m$ defined by $$\eta(i) = m + 1 - \sigma(i)$$ is a bijection satisfying $\operatorname{Opp}(\operatorname{MD}(m)) = \operatorname{DM}(m)$ and $\operatorname{Opp}(\operatorname{DM}(m)) = \operatorname{MD}(m)$. Verify that if $\sigma \in \Sigma_m$ and if $i, j$ are elements of $\{1, \ldots, m\}$ satisfying $\sigma(j) > \sigma(i)$, $$\sigma(j) - \sigma(i) = 1 + \operatorname{Card}\{k \in \Delta_m \mid \sigma(i) < \sigma(k) < \sigma(j)\}$$