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, Wa...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 Schr¨oder 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)-Schr¨oder 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.展开更多
基金The NSF(11601020,11501014)of ChinaOrganization Department of Beijing Municipal Committee(2013D005003000012)Science and Technology Innovation Platform-Business Project 2017(PXM2017_014213_000022)
文摘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 Schr¨oder 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)-Schr¨oder 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.
