Year: 2026
Type: preprint
Abstract: Given a finite preorder $\tau$, we study how its order relation determines the $h^*$-polynomial of the associated polar preorder polytope. Let $\mathcal{R}_{\tau}^{\vee}$ be the convex hull of the negative standard basis vectors and the indicator vectors of nonempty order ideals, and write $h_\tau^*(t)=h^*(\mathcal{R}_{\tau}^{\vee},t)$. Athanasiadis and Chapoton conjectured that this enumerative invariant takes ordinal sums to products:\[h_{\tau_1\oplus\tau_2}^*(t)=h_{\tau_1}^*(t)\,h_{\tau_2}^*(t).\]We prove the conjecture for arbitrary finite preorders, including those with nontrivial equivalence classes. More precisely, we construct a positively homogeneous piecewise unimodular shear from the free sum $\mathcal{R}_{\tau_1}^{\vee}\oplus\mathcal{R}_{\tau_2}^{\vee}$ onto $\mathcal{R}_{\tau_1\oplus\tau_2}^{\vee}$, giving lattice-point bijections at every nonnegative real dilation scale. The key is a formula for the Minkowski functional in terms of the least nonnegative order-reversing majorant: after the shear, the functional becomes the sum of the two factor functionals. Together with the Ehrhart product formula for free sums, this proves the conjecture and the corresponding product formula for normalized volume.
Source: arXiv (Cornell University)
Author: Luo, Congyi
Institution: Fudan University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213645933