Exact generating function for 2-convex polygons

TL;DR

This paper derives an exact generating function for 2-convex polygons, proving a conjecture and developing tools that could extend to polygons with higher concavity indices.

Contribution

It provides the first exact generating function for 2-convex polygons and introduces methods applicable to polygons with larger concavity indices.

Findings

Conjectured isotropic generating function for m=2 confirmed.

Derived anisotropic generating function for 2-convex polygons.

Developed tools for studying polygons with 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 their perimeter. We first describe how we conjectured the (isotropic) generating function for the case $m=2$ using a numerical procedure based on series expansions. We then proceed to prove this result for the more general case of the full anisotropic generating function, in which steps in the $x$ and $y$ direction are distinguished. In so doing, we develop tools that would allow for the case $m > 2$ to be studied. %In our proof 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.