Uniform partition extensions, a generating functions perspective

Abstract

In this paper, a bivariate generating function $CF(x,y) = \frac{{f(x) - yf(xy)}} {{1 - y}}$ is investigated, where f(x) = Σ _n⩾0 f _n x ⁿ is a generating function satisfying the functional equation f(x) = 1 + Σ ^r _j=1 Σ ^m _i=j−1 a _ij x ⁱ f(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.