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

TL;DR

This paper explores bijections and enumerations of generalized Motzkin paths avoiding certain patterns, linking them to Schr"oder, Dyck, and Motzkin paths, and analyzing specific statistics with connections to Riordan arrays.

Contribution

It introduces new bijections between pattern-avoiding G-Motzkin paths and classical lattice paths, and provides enumeration formulas for path statistics related to Riordan arrays.

Findings

Bijections between $ extbf{uvu}$-avoiding G-Motzkin paths and Schr"oder/Dyck paths.

Enumeration formulas for steps and points statistics in pattern-avoiding paths.

Connections established between path statistics and Riordan arrays.

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 $(a,b)$-Schr\"{o}der paths as well as the set of $(a+b,b)$-Dyck paths of length $2n$, between the set 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 $(a,b)$-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.