在德兰诺伊没有高峰和低谷的小路上
On Delannoy paths without peaks and valleys
摘要：
如果一条格路径的每一步都属于$\left\{N, E, D\right\}$，则该格路径称为\emph{Delannoy}，其中有$N=(0,1)$、$E=(1,0)$和$D=(1,1)$步。\emph{Peak}、\emph{valley}和\emph{deep valley}分别表示晶格路径上的$NE$、$EN$和$EENN$。本文找到了$\mathcal{P}_{n,m}(NE, EN)$与${\mathcal{P}_{n,m}}(D, EENN)$的一个特定子集之间的一个双射，其中$\mathcal{P}_{n,m}(NE, EN)$是从原点到点$(n,m)$的无峰无谷的Delannoy路的集合，而${\mathcal{P}_{n,m}}(D, EENN)$是从原点到点$(n,m)$的无对角台阶无深谷的Delannoy格路的集合.对于正整数$k$，我们还计算了限制区域$\left\{ (x,y) \in \mathbb{Z}^2 : y \ge k x \right\}$上无峰无谷的Delannoy路的个数.
英文摘要：
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 $k$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
