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A generalized Motzkin path, called G-Motzkin path for short, of length $n$ isa lattice path from $(0, 0)$ to $(n, 0)$ in the first quadrant of the XOY-planethat 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 $c$. In thispaper, we first give bijections between the set of $\mathbf{uvu}$-avoiding$(a,b,b^2)$-G-Motzkin paths of length $n$ and the set of $(a,b)$-Schr\"{o}derpaths as well as the set of $(a+b,b)$-Dyck paths of length $2n$, between theset of $\{\mathbf{uvu, uu}\}$-avoiding $(a,b,b^2)$-G-Motzkin paths of length$n$ and the set of $(a+b,ab)$-Motzkin paths of length $n$, between the set of$\{\mathbf{uvu,uu}\}$-avoiding $(a,b,b^2)$-G-Motzkin paths of length $n+1$beginning with an $\mathbf{h}$-step weighted by $a$ and the set of $(a,b)$-Dyckpaths of length $2n+2$. In the last section, we focus on the enumeration ofstatistics "number of $\mathbf{z}$-steps" for $\mathbf{z}\in \{\mathbf{u},\mathbf{h}, \mathbf{v}, \mathbf{d}\}$ and "number of points" at given level in$\mathbf{uvu}$-avoiding G-Motzkin paths. These counting results are linked withRiordan arrays.

The $\mathbf{uvu}$-avoiding $(a,b,c)$-Generalized Motzkin paths with vertical steps: bijections and statistic enumerations