A bijection between the sets of $(a,b,b^2)$-Generalized Motzkin paths avoiding $\mathbf{uvv}$-patterns and $\mathbf{uvu}$-patterns

Abstract

A generalized Motzkin path, called G-Motzkin path for short, of length $n$ is a lattice path from $(0,0)$ to $(n,0)$ in the first quadrant of the XOY-plane that consists of up steps $\mathbf{u}=(1,1)$, down steps $\mathbf{d}=(1,-1)$, horizontal steps $\mathbf{h}=(1,0)$ and vertical steps $\mathbf{v}=(0,-1)$. An $(a,b,c)$-G-Motzkin path is a weighted G-Motzkin path such that the $\mathbf{u}$-steps, $\mathbf{h}$-steps, $\mathbf{v}$-steps and $\mathbf{d}$-steps are weighted respectively by $1,a,b$ and $c$. Let $\ddot{I}„$ be a word on $\{\mathbf{u},\mathbf{d},\mathbf{v},\mathbf{d}\}$, denoted by ${\mathcal{G}}_n^{\ddot{I}}„(a,b,c)$ the set of $\ddot{I}„$-avoiding $(a,b,c)$-G-Motzkin paths of length $n$ for a pattern $\ddot{I}„$. In this paper, we consider the $\mathbf{uvv}$-avoiding $(a,b,c)$-G-Motzkin paths and provide a direct bijection $\ddot{I}ƒ$ between ${\mathcal{G}}_n^{\mathbf{uvv}}(a,b,b^2)$ and ${\mathcal{G}}_n^{\mathbf{uvu}}(a,b,b^2)$. Finally, the set of fixed points of $\ddot{I}ƒ$ is also described and counted.

11pages,2 figures. arXiv admin note: substantial text overlap with arXiv:2201.09236