论文标题
在没有峰值和山谷的Delannoy路径上
On Delannoy paths without peaks and valleys
论文作者
论文摘要
如果其每个步骤属于$ \ left \ {n，e，d \ right \} $，则称为晶格路径，其中$ n =（0,1）$，$ e =（1,0）$，和$ d =（1,1）$ steps。 \ emph {peak}，\ emph {valley}和\ emph {deep valley}的含义$ ne $，$ en $和$ eenn $分别在晶格路径上。 在本文中，我们发现$ \ Mathcal {p} _ {n，m}（ne，en）$与$ {\ Mathcal {p} _ {p} _ {n，m}}（d，d，eenn）$的特定子集与特定子集（$ {p} _ $（n，m）$没有峰值和山谷，以及$ {\ Mathcal {p} _ {n，m}}}（d，eenn）$是从原点到点$（n，m）$的Delannoy晶格路径的集合，而无需对角线和深valleys。我们还列举了在受限区域上没有峰值和山谷的Delannoy路径$ \ left \ {（x，y）\ in \ mathbb {z}^2：y \ ge k x \ x \ right \} $，用于正integer $ k $。
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 $k$.
