Interior vertices and edges in bargraphs

In this paper, we consider two statistics on bargraphs, which are defined to be lattice paths in the first quadrant, starting at the origin and ending upon first return to the x-axis. Each bargraph is represented as a sequence of columns π1π2 . . . πm such that column k contains πk cells. First we enumerate interior vertices, where naturally, interior vertex is a vertex that belongs to exactly four cells of bargraphs. Then we enumerate d-edges - edges that contain d
interior vertices. More precisely, we find the generating function for the number of bargraphs with n cells and m columns according: to interior vertices and according to horizontal (vertical) d-edges. In addition we consider several special cases in detail, where we obtain asymptotic results for total number of statistics under consideration.