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arxiv: 2605.26916 · v1 · pith:YDWOBNWP · submitted 2026-05-26 · math.CO

Polytopes and posets associated to preorders

Fr\'ed\'eric Chapoton (IRMA) , Christos A. Athanasiadis This is my paper

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classification math.CO
keywords preorder polytopeslattice polytopesEhrhart polynomialszeta polynomialsposets of lattice pointsarbor polytopesnormalized volumeh*-polynomials
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The pith

Preorder polytopes from preorders on finite sets are lattice polytopes with a duality linking Ehrhart polynomials to zeta polynomials of their lattice point posets.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors introduce preorder polytopes defined from preorders on finite sets. These generalize arbor polytopes and turn out to be lattice polytopes. A duality is proven that relates the Ehrhart polynomial of the polytope to the zeta polynomial of the poset of its lattice points. This leads to proven formulas for the Ehrhart polynomial, the h*-polynomial, and a combinatorial interpretation of the normalized volume, while a combinatorial meaning for the h*-polynomial is conjectured.

Core claim

Preorder polytopes, defined from preorders on finite sets, are lattice polytopes satisfying a duality that relates their Ehrhart polynomials to the zeta polynomials of their posets of lattice points. The normalized volume has a combinatorial interpretation, and formulas for the Ehrhart and h*-polynomials are given, with a conjecture on the h* interpretation. These results extend previous work on arbor polytopes to the broader setting of preorders.

What carries the argument

The preorder polytope constructed from a preorder on a finite set and the poset formed by its lattice points, which together enable the duality between Ehrhart and zeta polynomials.

Load-bearing premise

The specific construction of the preorder polytope from a preorder on a finite set produces a poset of lattice points whose zeta polynomial relates to the Ehrhart polynomial through the claimed duality.

What would settle it

Computing the Ehrhart polynomial and the zeta polynomial for the poset of lattice points of a small preorder polytope, such as for a preorder with three elements, and checking if they satisfy the duality relation.

Figures

Figures reproduced from arXiv: 2605.26916 by Christos A. Athanasiadis, Fr\'ed\'eric Chapoton (IRMA).

Figure 1
Figure 1. Figure 1: The Hasse diagram of a preorder on the set {a, b, c, d, e}. This paper generalizes arbor polytopes by replacing arbors on finite sets by arbitrary pre￾orders. A preorder on a finite set E can be viewed as a partial order on the set of blocks of a partition of E, called vertices (see Section 2.2). The example shown in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example of bipartite graph Gτ . 4.1. Ehrhart-Zeta duality. Let us define a bipartite graph Gτ with n + 1 left and n + 1 right vertices as follows. There are vertices ℓe on the left side and re on the right side, for e ∈ E ⊔ {0} [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
read the original abstract

Preorder polytopes, defined from preorders on finite sets, are introduced and studied from a lattice point enumeration point of view. They naturally generalize arbor polytopes, recently introduced and studied by the second named author. Preorder polytopes are shown to be lattice polytopes which satisfy a certain duality relating their Ehrhart polynomials with the zeta polynomials of their posets of lattice points. A combinatorial interpretation of the normalized volume of a preorder polytope is proven, together with formulas for the Ehrhart polynomial and the $h^\ast$-polynomial, and a combinatorial interpretation of the latter is conjectured. Several conjectures and results on the lattice point enumeration of arbor polytopes are generalized to preorder polytopes, new conjectures are proposed and new interesting examples of preorder polytopes are studied.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces preorder polytopes associated to preorders on finite sets, generalizing arbor polytopes. It proves that these polytopes are lattice polytopes satisfying a duality relating their Ehrhart polynomials to the zeta polynomials of the posets formed by their lattice points. A combinatorial interpretation of the normalized volume is established, explicit formulas are derived for the Ehrhart polynomial and the h*-polynomial, and a combinatorial interpretation of the h*-polynomial is conjectured. Several results and conjectures from the arbor polytope literature are generalized, with additional conjectures proposed and new examples examined.

Significance. If the central claims hold, the work supplies a broad new family of lattice polytopes in which Ehrhart theory is explicitly connected to poset combinatorics via an order-preserving bijection between lattice points and chains. The resulting volume formula (via maximal chains) and the substitution of known zeta polynomials into the Ehrhart series constitute concrete combinatorial contributions that extend the arbor-polytope results without additional restrictions.

minor comments (2)
  1. [Introduction] The abstract states that the duality, volume interpretation, and polynomial formulas are proven, yet the introduction would benefit from an explicit statement of the order-preserving bijection used to equate the Ehrhart series with the zeta generating function.
  2. [Section 2] A small concrete example (e.g., the preorder polytope for a two-element chain or antichain) placed immediately after the definition would help readers verify the lattice-point poset construction before the general proofs.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of the manuscript. The recommendation for minor revision is noted, though no specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivations rely on explicit bijections and standard identities

full rationale

The paper defines preorder polytopes from preorders on finite sets and derives the Ehrhart-zeta duality via an explicit order-preserving bijection between lattice points and chains in the associated poset, combined with the standard Ehrhart series generating-function identity. Normalized volume follows by evaluating at t=1 and counting maximal chains. Ehrhart and h*-polynomial formulas substitute the known zeta polynomial of the chain poset into the general duality. All steps are carried out directly from the definitions for arbitrary finite preorders, without reducing to fitted parameters, self-definitional loops, or load-bearing self-citations. Generalizations from prior arbor polytope work by one author are non-load-bearing extensions of the same explicit constructions. The central claims remain independent of their inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper introduces new combinatorial objects but relies on standard background from order theory and polytope theory without introducing fitted parameters or ad hoc axioms.

axioms (1)
  • standard math Standard definitions and properties of preorders, posets, lattice polytopes, Ehrhart polynomials, and zeta polynomials hold.
    The abstract invokes these established concepts to define and study the new polytopes.
invented entities (1)
  • preorder polytope no independent evidence
    purpose: To generalize arbor polytopes and enable study of lattice point enumeration and duality properties.
    New object constructed in the paper from preorders on finite sets.

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