Peaks in superdiagonal bargraphs

A bargraph is a first quadrant, non-intersecting lattice path consisting of up, down and horizontal steps. Superdiagonal bargraphs are bargraphs in which all horizontal steps (except possibly for the endpoints) are above the line y = x. Generating functions for the total number of valleys in superdiagonal bargraphs are found. Attention is then turned to peaks and generating functions for the width of peaks as well as the position of the first peak are found. This is done with respect to the parameters semi-perimeter and also total length. Asymptotics for the various statistics are found.