论文标题
晶格路径上的组合学
Combinatorics on lattice paths in strips
论文作者
论文摘要
For lattice paths in strips which begin at $(0,0)$ and have only up steps $U: (i,j) \rightarrow (i+1,j+1)$ and down steps $D: (i,j)\rightarrow (i+1,j-1)$, let $A_{n,k}$ denote the set of paths of length $n$ which start at $(0,0)$, end on heights $0$ or $ -1 $，并包含在条款$ - \ lfloor \ frac {k+1} {2} {2} \ rfloor \ rfloor \ leq y \ leq y \ leq \ leq \ lfloor \ frac {k} {k} {2} {2} \ rfloor width $ k $的$ k $，k $ n，k $ k $ n，k} $ n， $（0,0）$，并包含在Strip $ 0 \ leq y \ leq K $中。我们在$ a_ {n，k} $和$ b_ {n，k} $之间建立了两者。 还讨论了这两组子集的生成功能。此外，通过将路径转换为两种类型的树，我们在$ a_ {n，3} $和$ b_ {n，3} $之间提供了另一种两者。
For lattice paths in strips which begin at $(0,0)$ and have only up steps $U: (i,j) \rightarrow (i+1,j+1)$ and down steps $D: (i,j)\rightarrow (i+1,j-1)$, let $A_{n,k}$ denote the set of paths of length $n$ which start at $(0,0)$, end on heights $0$ or $-1$, and are contained in the strip $-\lfloor\frac{k+1}{2}\rfloor \leq y \leq \lfloor\frac{k}{2}\rfloor$ of width $k$, and let $B_{n,k}$ denote the set of paths of length $n$ which start at $(0,0)$ and are contained in the strip $0 \leq y \leq k$. We establish a bijection between $A_{n,k}$ and $B_{n,k}$. The generating functions for the subsets of these two sets are discussed as well. Furthermore, we provide another bijection between $A_{n,3}$ and $B_{n,3}$ by translating the paths to two types of trees.
