Volume 33, Issue 2
Two Bijections on Weighted Motzkin Paths

Zhongjin Chen & Shuo Zhao

Commun. Math. Res., 33 (2017), pp. 149-159.

Published online: 2019-11

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  • 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. 

  • Keywords

underdiagonal lattice path, (2,2)-Motzkin path, $k$-Motzkin path, ($k$−2)-Schröder path

  • AMS Subject Headings

05C38, 05A19

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

chenzjst@163.com (Zhongjin Chen)

  • BibTex
  • RIS
  • TXT
@Article{CMR-33-149, author = {Chen , Zhongjin and Zhao , Shuo}, title = {Two Bijections on Weighted Motzkin Paths}, journal = {Communications in Mathematical Research }, year = {2019}, volume = {33}, number = {2}, pages = {149--159}, 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. 

}, 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} }
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 -

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. 

Zhongjin Chen & Shuo Zhao. (2019). Two Bijections on Weighted Motzkin Paths. Communications in Mathematical Research . 33 (2). 149-159. doi:10.13447/j.1674-5647.2017.02.07
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