Large chirotopes with computable numbers of triangulations

TL;DR

This paper explores decomposition techniques for chirotopes, a combinatorial abstraction of planar point sets, and applies these methods to accurately estimate the number of triangulations supported by specific point configurations.

Contribution

It generalizes existing sum operations for chirotopes and derives a precise asymptotic estimate for triangulations of the double circle configuration.

Findings

Generalized sum operations for chirotopes.

Derived asymptotic estimate for triangulations of the double circle.

Applied kernel method to solve functional equations.

Abstract

Chirotopes are a common combinatorial abstraction of (planar) point sets. In this paper we investigate decomposition methods for chirotopes, and their application to the problem of counting the number of triangulations supported by a given planar point set. In particular, we generalize the convex and concave sums operations defined by Rutschmann and Wettstein for a particular family of chirotopes (which they call chains), and obtain a precise asymptotic estimate for the number of triangulations of the double circle, using a functional equation and the kernel method.