Walls in bargraphs

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 \(r \geq 1\). We then derive asymptotic estimates for the mean number of such walls.