Restricted bargraphs and unimodal compositions

Abstract

In this paper, we present a study on polyominoes, which are polygons created by connecting unit squares along their edges. Specifically, we focus on a related concept called a bargraph, which is a path on a lattice in $Z_{\ge 0}\times Z_{\ge 0}$ traced along the boundaries of a column convex polyomino where the lower edge is on the x-axis. To explore new variations of bargraphs, we introduce the notion of non-decreasing bargraphs, which incorporate an additional restriction concerning the valleys within the path. We establish intriguing connections between these novel objects and unimodal compositions. To facilitate our analysis, we employ generating functions, including q-series, as well as various closed formulas. These tools enable us to enumerate the different types of bargraphs based on their semi-perimeter, area, and the number of peaks. Furthermore, we provide combinatorial justifications for some of the derived closed formulas.