Uniform partition extensions,a generating functions perspective

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.