The height and width of bargraphs

A bargraph is a lattice path in N02 with three allowed steps: the up step u=(0,1)u=(0,1), the down step d=(0,−1)d=(0,−1) and the horizontal step h=(1,0)h=(1,0). It starts at the origin with an up step and terminates as soon as it intersects the xx-axis again. A down step cannot follow an up step and vice versa. The height   of a bargraph is the maximum yy coordinate attained by the graph. The width is the horizontal distance from the origin till the end. For bargraphs of fixed semi-perimeter nn we find the generating functions for the total height and the total width and hence find asymptotic estimates for the average height and the average width. Our methodology makes use of a bijection between bargraphs and uudduudd-avoiding Dyck paths.