Families of m-convex polygons: m = 2

Abstract

Polygons are described as almost-convex if their perimeter differs from the perimeter of their minimum bounding rectangle by twice their `concavity index', $m$. Such polygons are called \emph{$m$-convex} polygons and are characterised by having up to $m$ indentations in the side. We use a `divide and conquer' approach, factorising 2-convex polygons by extending a line along the base of its indents. We then use the inclusion-exclusion principle, the Hadamard product and extensions to known methods to derive the generating functions for each case.