Counting Number of Triangulations of Point Sets: Reinterpreting and Generalizing the Triangulation Polynomials
Hong Duc Bui
TL;DR
This work introduces a unifying algebraic framework for counting triangulations of point sets by connecting two prior polynomial formalisms and extending them from chains to near-edges. It develops univariate and bivariate triangulation polynomials, with multiplicative operators that enable modular composition via gluing, convex/concave sums, and flipping, allowing efficient computation of triangulations for several near-edge configurations. The paper also derives closed-form relations between the $t$- and $m$-series, introduces hat-operators for series extension, and provides numerical experiments that probe growth rates and identify near-edges that maximize triangulation counts, notably via Koch-like constructions. Overall, the framework clarifies the algebraic structure behind triangulation counting and offers practical, polynomial-time methods for counting in broader families, while suggesting that existing exponential bounds are close to optimal in many cases.
Abstract
We describe a framework that unifies the two types of polynomials introduced respectively by Bacher and Mouton and by Rutschmann and Wettstein to analyze the number of triangulations of point sets. Using this insight, we generalize the triangulation polynomials of chains to a wider class of near-edges, enabling efficient computation of the number of triangulations of certain families of point sets. We use the framework to try to improve the result in Rutschmann and Wettstein without success, suggesting that their result is close to optimal.