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 $\tau$ be a word on ${\mathbf{u}, \mathbf{d}, \mathbf{v}, \mathbf{d}}$, denoted by $\mathcal{G}_n^{\tau}(a,b,c)$ the set of $\tau$-avoiding $(a,b,c)$-G-Motzkin paths of length $n$ for a pattern $\tau$. In this paper, we consider the $\mathbf{uvv}$-avoiding $(a,b,c)$-G-Motzkin paths and provide a direct bijection $\sigma$ 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 $\sigma$ is also described and counted.