Combinatorics on lattice paths in strips

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.

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