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 $\sigma(2) - \sigma(1) = s+1, \quad n+2-\sigma(2) = k$. 2. Under what condition (necessary and sufficient) on $n$ and $k$ is the set $\mathcal{B}(n, k)$ non-empty? Under what condition (necessary and sufficient) on $n$, $s$ and $k$ is the set $\mathcal{C}(n, s, k)$ non-empty?
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 $\sigma(2) - \sigma(1) = s+1, \quad n+2-\sigma(2) = k$.
2. Under what condition (necessary and sufficient) on $n$ and $k$ is the set $\mathcal{B}(n, k)$ non-empty? Under what condition (necessary and sufficient) on $n$, $s$ and $k$ is the set $\mathcal{C}(n, s, k)$ non-empty?