The Number of Polytope Trees and Beyond
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.
Related
- On the limiting distribution of the number of improper edges for random trees
- Maximum number of spanning trees in bipartite graphs with a given diameter
- The number of Laplacian eigenvalues of trees less than one
- The packing and covering number of trees are local
- A Turán-type extremal problem for the number of spanning trees in C4-free graphs
BibTeX
@misc{aliev2026number,
title = {{The Number of Polytope Trees and Beyond}},
author = {Ali Aliev and Natalia Amburg},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33344},
eprint = {2610.33344},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33344},
}
RIS
TY - UNPB
TI - The Number of Polytope Trees and Beyond
AU - Ali Aliev
AU - Natalia Amburg
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.33344
UR - https://arxiv.org/abs/2610.33344
AB - 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.
ER -