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Random question for #mathstodon (the server and community generally):
Given a number n of points in a d dimensional space, we can think of the Delaunay triangulation of a configuration of those points as an adjacency matrix. I'm curious if anyone knows how many distinct configurations you can have: where two configurations are distinct if their adjacency matrices are not equal.
I haven't been able to work out a solution myself and most of my searches don't get me much.
@4sphere In the case of d=2, this amounts to finding how many distinct graphs arise as the 1-skeleton of a Delaunay cellulation? In turn, I think this can be reduced to determining which graphs are inscribable in the sphere. Adding a point at infinity connected to all of the outermost points of the Daulaunay graph will give an inscribable polyhedron, thought of as the convex hull of ideal points in the upper half-space model of hyperbolic space.
@4sphere There is an algorithm to characterize inscribable polyhedra, but it might be hard to turn this in to an enumeration. https://mathscinet.ams.org/mathscinet-getitem?mr=1149872.
@4sphere Looking at the papers citing that one, I found another that gives a characterization of planar Delaunay triangulations. https://mathscinet.ams.org/mathscinet-getitem?mr=3746818