Lattice paths moments by cut and paste

In the coordinate plane consider those lattice paths whose step types consist of (1, 1), (1,−1), and perhaps one or more horizontal steps. For the set of such paths running from (0, 0) to (n+ 2, 0) and remaining strictly elevated above the horizontal axis elsewhere, we define a zeroth moment (cardinality), a first moment (essentially, the total area), and a second moment, each in terms of the ordinates of the lattice points traced by the paths. We then establish a bijection relating these moments to the cardinalities of sets of selected marked unrestricted paths running from (0, 0) to (n, 0). Roughly, this bijection acts by cutting each elevated path into well-defined subpaths and then pasting the subpaths together in a specified order to form an unrestricted path.