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

TL;DR

This paper establishes a bijection between two classes of pattern-avoiding generalized Motzkin paths with specific weights, providing insights into their combinatorial structure and fixed points.

Contribution

It introduces a direct bijection between $ extbf{uvv}$-avoiding and $ extbf{uvu}$-avoiding $(a,b,b^2)$-G-Motzkin paths, revealing their structural relationship.

Findings

A bijection between the two classes of paths is constructed.

The fixed points of the bijection are characterized and enumerated.

The study enhances understanding of pattern avoidance in weighted lattice paths.

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.