Moments, Narayana numbers, and the cut and paste for lattice paths

Let U(n) denote the set of unrestricted lattice paths that run from (0,0) to (n,0) with permitted steps (1,1), (1,-1), and perhaps a horizontal step. Let E(n+2) denote the set of paths in U(n+2) that run strictly above the horizontal axis except initially and finally. First we review the cut-and-paste bijection which relates points under paths of E(n+2) to points on paths of U(n). We apply it to obtain area and enumeration results for paths, some involving the Narayana distribution. We extend the cut-and-paste bijection to a formula relating factorial moments for the paths of E(n+2) to factorial moments for the paths of U(n).