Large chirotopes with computable numbers of triangulations
This work addresses a combinatorial geometry problem for researchers in computational geometry, providing incremental advances by extending known methods to more general chirotopes.
The paper tackled the problem of counting triangulations of planar point sets by generalizing decomposition methods for chirotopes, specifically extending convex and concave sums operations from chains to broader families, and obtained a precise asymptotic estimate for the number of triangulations of the double circle using functional equations and the kernel method.
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.