  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 $k$.
