TY - JOUR
T1 - Two Bijections on Weighted Motzkin Paths
AU - Chen , Zhongjin
AU - Zhao , Shuo
JO - Communications in Mathematical Research 
VL - 2
SP - 149
EP - 159
PY - 2019
DA - 2019/11
SN - 33
DO - http://doi.org/10.13447/j.1674-5647.2017.02.07
UR - https://global-sci.org/intro/article_detail/cmr/13395.html
KW - underdiagonal lattice path, (2,2)-Motzkin path, $k$-Motzkin path, ($k$−2)-Schröder path
AB - <p style="text-align: justify;">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.&nbsp;</p>