The asymptotic number of zonotopes

Olivier Bodini; Théophile Buffière

Home/Archive/arXiv

arXiv · 2610.39900

The asymptotic number of zonotopes

Olivier Bodini·Théophile Buffière

View original source ↗Read source document ↗Export metadata ↓

Abstract

We provide a sharp estimate for the asymptotic number of lattice zonotopes inscribed in [0, n] d as n tends to infinity. Our estimate refines the logarithmic equivalent established by B{á}r{á}ny, Bureaux, and Lund when the sum of the generators of the zonotope is prescribed. As we shall see, the exponential part of our estimate is a polynomial of degree d in n 1/(d+1) , plus a term involving the non-trivial zeros of the Riemann zeta function. Our analysis is based on a correspondence between power sums over primitive vectors (vectors with coprime coordinates) and Eulerian polynomials. We also analyse some combinatorial properties of lattice zonotopes. In particular, we compute the asymptotic mean and variance of the diameter of the graph of a random lattice zonotope inscribed in [0, n] d as n goes to infinity, and show that this diameter satisfies a central limit theorem.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Olivier Bodini, Théophile Buffière. 2026-10-29. The asymptotic number of zonotopes. https://arxiv.org/abs/2610.39900

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

How to Tropicalize a non-Archimedean Lattice

We present a method for computing the tropicalization of a lattice over the valuation ring of a non-Archimedean field. Our construction relies on the entropy polynomial, a tropical polynomial whose coefficients are determined by the valuations of minors of a matrix representing the lattice. We show that the tropicalization of the lattice is the support of a polyhedral complex defined through this polynomial. Moreover, we prove that this tropicalization is a finitely generated tropical semimodule over the nonnegative tropical scalars and provide explicit generators in terms of the entropy vector of the lattice. Finally, we extend our construction to valuated bimatroids, establishing analogous results beyond the realizable setting.

math.CO↗

arXiv2026

Arboreal Networks and Ultrametrics

Ultrametrics are an important class of distances used in applications such as phylogenetics, clustering and classification theory. Ultrametrics are essentially distances that can be represented by an edge-weighted rooted tree so that all of the distances in the tree from the root to any leaf of the tree are equal. In this paper, we introduce a generalization of ultrametrics called arboreal ultrametrics which have applications in phylogenetics and also arise in the theory of distance-hereditary graphs. These are partial distances, that is distances that are not necessarily defined for every pair of elements in the groundset, that can be represented by an ultrametric arboreal network, that is, an edge-weighted rooted network whose underlying graph is a tree. As with ultrametrics all of the distances in the ultrametric arboreal network from any root to any leaf below it are are equal but, in contrast, the network may have more than one root. In our two main results we characterize when a partial distance is an arboreal ultrametric as well as proving that, somewhat surprisingly, given any unrooted edge-weighted phylogenetic tree there is a necessarily unique way to insert roots into this tree so as to obtain an arboreal ultrametric.

math.CO↗

arXiv2026

Coloring t-perfect graphs with fewer colors

Recently, Chudnovsky, Cook, Davies, Oum, and Tan obtained the first finite bound on the chromatic number of t-perfect graphs, showing that they are 199053-colorable. We improve this bound to 130 by refining their proof. The original proof establishes that every graph with large odd girth and large chromatic number contains a certain structure called an r-arithmetic rope, and that its existence in a certain leveling of a graph with large odd girth would imply an odd wheel as a t-minor, a known obstruction to t-perfectness. While their technique requires a lower bound on the chromatic number that is exponential in r, we show that the existence of an r-arithmetic rope can already be guaranteed under a linear bound. Using a slightly weakened notion of arithmetic ropes allows us to reduce the bound even further.

math.CO↗