Families of m-convex polygons: m = 2

TL;DR

This paper introduces a combinatorial approach to enumerate m-convex polygons, focusing on 2-convex polygons, by extending existing methods and deriving their generating functions.

Contribution

It develops a divide and conquer method for 2-convex polygons and derives their generating functions using advanced combinatorial techniques.

Findings

Derived generating functions for 2-convex polygons

Extended known enumeration methods to m-convex polygons

Provided a new combinatorial framework for polygon classification

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.