Walls in bargraphs

Abstract

Bargraphs are lattice paths in ${\mathbb{N}}_0^2$ with three allowed types of steps: up $(0,1)$, down $(0,-1)$, and horizontal $(1,0)$. They start at the origin with an up step and terminate immediately upon return to the $x$-axis. A wall of size $r$ is a maximal sequence of $r$ adjacent up steps. In this paper, we develop the generating function for the total number of walls of fixed size $r\ge 1$. We then derive asymptotic estimates for the mean number of such walls.