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

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 $c$. In this paper, 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 $(a,b)$-Schröder paths as well as the set of $(a+b,b)$-Dyck paths of length $2n$, between the set of $\{\mathbf{uvu},\mathbf{uu}\}$-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 $\{\mathbf{uvu},\mathbf{uu}\}$-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 $(a,b)$-Dyck paths of length $2n+2$. In the last section, we focus on the enumeration of statistics "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 with Riordan arrays.

18 pages, 4 figures