Prefixes of bargraph paths.

We extend the notion of bargraphs as first quadrant, semi-perimeter defined lattice paths to that of a bargraph prefix where we relax the bargraph defining parameters and allow the bargraph to extend into negative territory. A bargraph prefix is any initial section of a bargraph. Generating functions for the prefixes which separately track the number of up, down and horizontal steps are found. The asymptotics for the average height of the last prefix step in the different bargraph extensions is given. [ABSTRACT FROM AUTHOR]