On Delannoy paths without peaks and valleys

Abstract

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 $\left\{(x,y)\in {\mathbb{Z}}^2:y\ge kx\right\}$ for a positive integer $k$.

13 pages, 2 figures