Levels in bargraphs - Blecher, Aubrey (Ars mathematica contemporanea, 2015)
Bargraphs are lattice paths in ?$\mathbb{N}_0^2$?, which start at the origin and terminate immediately upon return to the ?$x$?-axis. The allowed steps are the up step ?$(0,1)$?, the down step ?$(0,-1)$? and the horizontal step ?$(1,0)$?. The first step is an up step and the horizontal steps must all lie above the ?$x$?-axis. An up step cannot follow a down step and vice versa. In this paper we consider levels, which are maximal sequences of two or more adjacent horizontal steps. We find the generating functions that count the total number of levels, the leftmost ?$x$?-coordinate and the height of the first level and obtain the generating function for the mean of these parameters. Finally, we obtain the asymptotics of these means as the length of the path tends to infinity.
Keywords: asimptotično vedenje; asymptotics; bargraphs; generating functions; levels; rodovne funkcije; stopničasti grafi; terase