With E. Pergola, R. Pinzani, and S. Rinaldi,
Lattice Path Moments by Cut and Paste,
{\it The Proceedings of the Conference on Formal Power Series and Algebraic
Combinatorics},\ University of Arizona, 2001.
In the coordinate plane consider
those lattice paths whose step types consists of (1,1), (1,-1), and
perhaps one or more horizontal steps. For
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, in terms of the ordinates of the
lattice points traced by its paths. We then establish a bijection
relating these moments to the cardinalities of sets of selected
lattice points on the 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.