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).