Counting the triangulations of a point set is #P-hard in general. But specific configurations — convex position, double circles, stacked constructions — admit exact enumeration through structural decomposition. The chirotope, a combinatorial abstraction that records the orientation of every triple of points, captures everything about a point set that triangulations can see.
For the double circle — points arranged on two concentric circles — convex and concave sum operations decompose the chirotope into simpler components whose triangulation counts satisfy functional equations (arXiv:2603.10251). The kernel method, a technique from analytic combinatorics, then extracts precise asymptotic estimates from these equations. The number of triangulations grows exponentially, and the growth rate is computable.
The structural insight: decomposition and counting are the same operation viewed from different directions. Breaking a chirotope into components generates functional equations; solving those equations counts the original object's triangulations. The combinatorial structure that enables decomposition is the same structure that enables enumeration. You cannot count without decomposing, and you cannot decompose without implicitly counting. The #P-hardness of the general problem is not that counting is hard — it is that decomposition is hard. Where decomposition is natural, counting follows.