论文标题
Gessel号码的注释
A Note on the Gessel Numbers
论文作者
论文摘要
Gessel编号$ P（N，R）$代表具有单位水平和垂直步骤从$（0,0）$到$（N+R，N+R-1）$的晶格路径的数量，这些步骤从未触及集合$ \ {（（x，x）\ in \ in \ in \ MathBB {z}^2：x：x：x \ geq r中的任何点。在本文中，我们使用组合论点来得出$ p（n，r）$和$ p（n-1，r+1）$之间的复发关系。另外，我们为$ p（n，r）$的知名封闭式公式提供了新的证明。此外，提出了针对Gessel数字的新组合解释。
The Gessel number $P(n,r)$ represents the number of lattice paths in a plane with unit horizontal and vertical steps from $(0,0)$ to $(n+r,n+r-1)$ that never touch any of the points from the set $\{(x,x)\in \mathbb{Z}^2: x \geq r\}$. In this paper, we use combinatorial arguments to derive a recurrence relation between $P(n,r)$ and $P(n-1,r+1)$. Also, we give a new proof for a well-known closed formula for $P(n,r)$. Moreover, a new combinatorial interpretation for the Gessel numbers is presented.
