Partial Motzkin paths with air pockets of the first kind avoiding peaks, valleys or double rises

TL;DR

This paper introduces a new class of Motzkin paths with air pockets, providing enumeration formulas, bijections with Dyck and Motzkin paths, and analyzing paths avoiding specific features like peaks, valleys, or double rises.

Contribution

It defines and enumerates MAP of the first kind with restrictions, using Riordan arrays and establishing bijections with other lattice paths.

Findings

Enumeration formulas for MAP with restrictions

Representation of results via Riordan arrays

Constructive bijections with Dyck and Motzkin paths

Abstract

Motzkin paths with air pockets (MAP) of the first kind are defined as a generalization of Dyck paths with air pockets. They are lattice paths in $\mathbb{N}^2$ starting at the origin made of steps $U=(1,1)$, $D_k=(1,-k)$, $k\geq 1$ and $H=(1,0)$, where two down-steps cannot be consecutive. We enumerate MAP and their prefixes avoiding peaks (resp. valleys, resp. double rise) according to the length, the type of the last step, and the height of its end-point. We express our results using Riordan arrays. Finally, we provide constructive bijections between these paths and restricted Dyck and Motzkin paths.