Commun. Math. Res., 33 (2017), pp. 149-159.
Published online: 2019-11
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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.
}, issn = {2707-8523}, doi = {https://doi.org/10.13447/j.1674-5647.2017.02.07}, url = {http://global-sci.org/intro/article_detail/cmr/13395.html} }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.