The area determined byunderdiagonal lattice

We use the \\rst passage decomposition" methodology to study the area between various kinds of underdiagonal lattice paths and the main diagonal. This area is important because it is connected to the number of inversions in permutations and to the internal path length in various types of trees. We obtain the generating functions for the total area of all the lattice paths from the origin to the point (n; n). Since this method also determines the number of these paths, we are able to obtain exact results for the average area.