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First page of The Number of Polytope Trees and Beyond

The Number of Polytope Trees and Beyond

Ali Aliev, Natalia Amburg

math.CO Oct 23, 2026 · v1 math-ph math.GT
formalization combinatorics
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TL;DR
Nearly all of the paper's enumerative results were formalized in Lean 4 with AI assistance.
Abstract
We study the symmetry-weighted enumeration of complexes formed by gluing copies of a regular polytope. For polytope trees and complexes obtained by attaching such trees to a fixed initial complex, we derive cut-and-join recurrences, differential equations for generating functions, explicit counting formulas, and asymptotic estimates. We then extend the method to complexes with cycles in their dual graphs, enumerating them by the numbers of polytopes and free facets. We derive recurrence relations and partial differential equations for the corresponding bivariate generating functions. In this setting, we obtain recurrences for connected configurations and explicit coefficient formulas for possibly disconnected configurations, both with and without a distinguished initial complex.
Problem

The paper counts complexes built by gluing copies of a regular polytope along facets, weighting each complex by the inverse size of its automorphism group. It treats polytope trees, trees attached to a fixed initial complex, and complexes whose dual graphs contain cycles.

Approach

Labelling external codimension-1 facets turns the problem into a cut-and-join decomposition, which yields recurrences for tetrahedron trees and general polytope trees. These recurrences are converted into differential equations for exponential generating functions, which are solved for explicit formulas and asymptotics. The method is extended to bivariate generating functions indexed by the number of polytopes and the cyclomatic number of the dual graph. Nearly the entire work was formalized in Lean 4 with AI assistance.

Figure 5: Visual explanation for recurrence.
Figure 7: Attaching a tetrahedron tree to an initial triangulation.
Results

The paper derives explicit counting formulas and asymptotic estimates for polytope trees and inductive complexes. It gives recurrences and PDEs for complexes with cycles, with explicit coefficient formulas for possibly disconnected configurations. The degenerate case h=2 (segments) is solved in closed form.