Combinatorics on lattice paths in strips

Abstract

For lattice paths in strips which begin at $(0,0)$ and have only up steps $U:(i,j)\to (i+1,j+1)$ and down steps $D:(i,j)\to (i+1,j-1)$, 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 $-\lfloor \frac{k+1}{2}\rfloor \le y\le \lfloor \frac{k}{2}\rfloor$ 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 $0\le y\le k$. 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.