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

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 XY-plane that consists of up steps $\mathrm{u}=(1, 1)$, horizontal steps $\mathrm{h}=(1, 0)$, vertical steps $\mathrm{v}=(0, -1)$ and down steps $\mathrm{d}=(1, -1)$. An $(a,b,c)$-G-Motzkin path is a weighted G-Motzkin path such that the $\mathrm{u}$-steps, $\mathrm{h}$-steps, $\mathrm{v}$-steps and $\mathrm{d}$-steps are weighted respectively by $1, a, b$ and $c$.Let $\tau$ be a word on $\{\mathrm{u}, \mathrm{h}, \mathrm{v}, \mathrm{d}\}$, denote 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 $\mathrm{uvv}$-avoiding $(a,b,c)$-G-Motzkin paths and provide a direct bijection $\sigma$ between $\mathcal{G}_n^{\mathrm{uvv}}(a,b,b^2)$ and $\mathcal{G}_n^{\mathrm{uvu}}(a,b,b^2)$. Finally, the set of fixed points of $\sigma$ is also described and counted.

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