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
In this paper, we study the class S of skew Motzkin paths, i.e., of those lattice paths that are in the first quadrat, which begin at the origin, end on the x-axis, consist of up steps U =（1, 1）,down steps D =（1,-1...In this paper, we study the class S of skew Motzkin paths, i.e., of those lattice paths that are in the first quadrat, which begin at the origin, end on the x-axis, consist of up steps U =（1, 1）,down steps D =（1,-1）, horizontal steps H =（1, 0）, and left steps L =（-1,-1）, and such that up steps never overlap with left steps. Let S;be the set of all skew Motzkin paths of length n and let 8;= |S;|. Firstly we derive a counting formula, a recurrence and a convolution formula for sequence{8;}n≥0. Then we present several involutions on S;and consider the number of their fixed points.Finally we consider the enumeration of some statistics on S;.展开更多
Bracketed words are basic structures both in mathematics （such as Rota-Baxter algebras） and mathematical physics （such as rooted trees） where the locations of the substructures are important. In this paper, we giv...Bracketed words are basic structures both in mathematics （such as Rota-Baxter algebras） and mathematical physics （such as rooted trees） where the locations of the substructures are important. In this paper, we give the classification of the relative locations of two bracketed subwords of a bracketed word in an operated semigroup into the separated, nested, and intersecting cases. We achieve this by establishing a correspondence between relative locations of bracketed words and those of words by applying the concept of Motzkin words which are the algebraic forms of Motzkin paths.展开更多
In this paper, a bivariate generating function CF(x, y) =f(x)-yf(xy)1-yis investigated, where f(x)= n 0fnxnis a generating function satisfying the functional equation f(x) = 1 + r j=1 m i=j-1aij xif(x)j.In particular,...In this paper, a bivariate generating function CF(x, y) =f(x)-yf(xy)1-yis investigated, where f(x)= n 0fnxnis a generating function satisfying the functional equation f(x) = 1 + r j=1 m i=j-1aij xif(x)j.In particular, we study lattice paths in which their end points are on the line y = 1. Rooted lattice paths are defined. It is proved that the function CF(x, y) is a generating function defined on some rooted lattice paths with end point on y = 1. So, by a simple and unified method, from the view of lattice paths, we obtain two combinatorial interpretations of this bivariate function and derive two uniform partitions on these rooted lattice 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.
基金Supported by National Natural Science Foundation of China(Grant No.11571150)
文摘In this paper, we study the class S of skew Motzkin paths, i.e., of those lattice paths that are in the first quadrat, which begin at the origin, end on the x-axis, consist of up steps U =（1, 1）,down steps D =（1,-1）, horizontal steps H =（1, 0）, and left steps L =（-1,-1）, and such that up steps never overlap with left steps. Let S;be the set of all skew Motzkin paths of length n and let 8;= |S;|. Firstly we derive a counting formula, a recurrence and a convolution formula for sequence{8;}n≥0. Then we present several involutions on S;and consider the number of their fixed points.Finally we consider the enumeration of some statistics on S;.
基金Acknowledgements The authors thank the Kavli Institute for Theoretical Physics China and the Morningside Center for Mathematics in Beijing for their hospitality and support. This work was supported by the National Natural Science Foundation of China （Grant Nos. 11201201, 11371178）, the Fundamental Research Funds for the Central Universities （lzujbky-2013-8）, and the National Science Foundation of US （Grant No. DMS 1001855）.
文摘Bracketed words are basic structures both in mathematics （such as Rota-Baxter algebras） and mathematical physics （such as rooted trees） where the locations of the substructures are important. In this paper, we give the classification of the relative locations of two bracketed subwords of a bracketed word in an operated semigroup into the separated, nested, and intersecting cases. We achieve this by establishing a correspondence between relative locations of bracketed words and those of words by applying the concept of Motzkin words which are the algebraic forms of Motzkin paths.
基金the "973" Project on Mathematical Mechanizationthe National Science Foundation, the Ministry of Education, and the Ministry of Science and Technology of China.
基金supported by National Natural Science Foundation of China(Grant No.11071163)Specialized Research Fund for the Doctoral Program of Higher Education(Grant No.20110073120068)Education Department of Henan Province(Grant No.14A110026)
文摘In this paper, a bivariate generating function CF(x, y) =f(x)-yf(xy)1-yis investigated, where f(x)= n 0fnxnis a generating function satisfying the functional equation f(x) = 1 + r j=1 m i=j-1aij xif(x)j.In particular, we study lattice paths in which their end points are on the line y = 1. Rooted lattice paths are defined. It is proved that the function CF(x, y) is a generating function defined on some rooted lattice paths with end point on y = 1. So, by a simple and unified method, from the view of lattice paths, we obtain two combinatorial interpretations of this bivariate function and derive two uniform partitions on these rooted lattice paths.
