Counting generalized Schr\"oder paths

A Schr\"oder path is a lattice path from $(0,0)$ to $(2n,0)$ with steps $(1,1)$, $(1,-1)$ and $(2,0)$ that never goes below the $x-$axis. A small Schr\"{o}der path is a Schr\"{o}der path with no $(2,0)$ steps on the $x-$axis. In this paper, a 3-variable generating function $R_L(x,y,z)$ is given for Schr\"{o}der paths and small Schr\"{o}der paths respectively. As corollaries, we obtain the generating functions for several kinds of generalized Schr\"{o}der paths counted according to the order in a unified way.