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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 = {Zhongjin and Chen and chenzjst@163.com and 5471 and School of Mathematics, Beijing Technology and Business University, Beijing, 100048 and Zhongjin Chen and Shuo and Zhao and and 5472 and School of Mathematics, Beijing Technology and Business University, Beijing, 100048 and Shuo Zhao}, 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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