Two Bijections on Weighted Motzkin Paths

Abstract

In this paper, we provide a bijection between the set of underdiagonal lattice paths of length $n$ and the set of (2,2)-Motzkin paths of length $n$. Besides, we generalize the bijection of Shapiro and Wang (Shapiro L W, Wang C J. A bijection between 3-Motzkin paths and Schröder paths with no peak at odd height. J. Integer Seq., 2009, 12: Article 09.3.2.) to a bijection between $k$-Motzkin paths and ($k$−2)-Schröder paths with no horizontal step at even height. It is interesting that the second bijection is a generalization of the well-known bijection between Dyck paths and 2-Motzkin paths.