Uniform partition extensions,a generating functions perspective  

作　　者：LI ShanHai MA Jun YEH YeongNan 

机构地区：[1]School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics [2]Department of Mathematics, Shanghai Jiao Tong University [3]Institute of Mathematics,Academia Sinica

出　　处：《Science China Mathematics》2015年第12期2655-2670,共16页中国科学：数学（英文版）

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)

摘　　要：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.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.

关 键 词：Chung-Feller theorem Dyck path Motzkin path Schr¨oder path 

分 类 号：O174[理学—数学]