On Delannoy paths without peaks and valleys

A lattice path is called \emph{Delannoy} if its every step belongs to $\left\{N, E, D\right\}$, where $N=(0,1)$, $E=(1,0)$, and $D=(1,1)$ steps. \emph{Peak}, \emph{valley}, and \emph{deep valley} mean $NE$, $EN$, and $EENN$ on the lattice path, respectively. In this paper, we find a bijection between $\mathcal{P}_{n,m}(NE, EN)$ and a specific subset of ${\mathcal{P}_{n,m}}(D, EENN)$, where $\mathcal{P}_{n,m}(NE, EN)$ is the set of Delannoy paths from the origin to the points $(n,m)$ without peaks and valleys and ${\mathcal{P}_{n,m}}(D, EENN)$ is the set of Delannoy lattice paths from the origin to the points $(n,m)$ without diagonal steps and deep valleys. We also enumerate the number of Delannoy paths without peaks and valleys on the restricted region $\left\{ (x,y) \in \mathbb{Z}^2 : y \ge k x \right\}$ for a positive integer $k$.