Exact generating function for 2-convex polygons
W. R. G. James, I. Jensen, A. J. Guttmann

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', . Such polygons are called \emph{-convex} polygons and are characterised by having up to indentations in their perimeter. We first describe how we conjectured the (isotropic) generating function for the case 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 and direction are distinguished. In so doing, we develop tools that would allow for the case 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…
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