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arxiv: 2607.00922 · v1 · pith:6VQ4SCBUnew · submitted 2026-07-01 · 🧮 math.CO

Order polytopes of generalized snake posets are h^*-real-rooted

Benjamin Braun , Aryaman Jal This is my paper

Pith reviewed 2026-07-02 10:34 UTC · model grok-4.3

classification 🧮 math.CO
keywords order polytopesgeneralized snake posetsh*-polynomialsreal-rooted polynomialsnon-nesting rook polynomialsEhrhart polynomialsflow polytopes
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The pith

Order polytopes of generalized snake posets have real-rooted Ehrhart h*-polynomials.

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

The paper proves that the Ehrhart h*-polynomials of order polytopes for generalized snake posets are real-rooted. This settles a conjecture about these polytopes, which are unimodularly equivalent to certain flow polytopes. The proof uses an identity that connects the h*-polynomials to non-nesting rook polynomials in a way that transfers real-rootedness. A sympathetic reader cares because real-rooted polynomials often encode positivity and combinatorial structure in counting problems for posets and polytopes.

Core claim

The Ehrhart h*-polynomials of the order polytopes for generalized snake posets are real-rooted. This is established by relating them to non-nesting rook polynomials via an identity or generating function that preserves the real-rooted property.

What carries the argument

The identity or generating function connecting these h*-polynomials to non-nesting rook polynomials, which preserves real-rootedness.

If this is right

  • The h*-polynomials have all roots real and negative.
  • The same real-rootedness holds for the equivalent strength-one flow polytopes.
  • Coefficients of these h*-polynomials admit combinatorial interpretations via non-nesting rook placements.
  • Real-rootedness transfers known inequalities and log-concavity properties from rook polynomials to these Ehrhart polynomials.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same linking technique could be tested on order polytopes of other narrow posets.
  • Real-rooted h*-polynomials may imply specific asymptotic distributions for the number of linear extensions in generalized snake posets.
  • The result suggests looking for similar rook polynomial connections in Ehrhart theory for width-two posets more broadly.

Load-bearing premise

The relation between the h*-polynomials and non-nesting rook polynomials preserves real-rootedness.

What would settle it

A generalized snake poset whose order polytope has an h*-polynomial containing at least one non-real complex root.

Figures

Figures reproduced from arXiv: 2607.00922 by Aryaman Jal, Benjamin Braun.

Figure 1
Figure 1. Figure 1: Generalized snake poset P for w = εRRLLLRLRR and the skew shape corresponding to P. The two chains give rise to the rows and columns of the skew shape. The cover relations ↗ and ↖ correspond respectively to the outer and inner corners of the skew shape. Proposition 2.5 (Obreschkoff, [Obr63, Satz 5.2]). Let f, g ∈ R[t] be non-zero polynomials with real roots. Then αf + βg is real-rooted for all α, β ∈ R if … view at source ↗
read the original abstract

Order polytopes for generalized snake posets were recently studied by von Bell et al. (2022), and are known to be unimodularly equivalent to strength-one flow polytopes for acyclic directed graphs strongly dual to generalized snake posets. Lee, Vindas-Mel\'endez, and Wang (2026) conjectured that the Ehrhart $h^*$-polynomials of these order polytopes are real-rooted. We prove this conjecture using a connection between these $h^*$-polynomials and non-nesting rook polynomials, which were recently introduced by Alexandersson and Jal (2024+) in connection with $P$-Eulerian polynomials for width two posets.

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 / 3 minor

Summary. The manuscript proves the conjecture of Lee, Vindas-Meléndez, and Wang that the Ehrhart h*-polynomials of order polytopes of generalized snake posets are real-rooted. The proof is obtained by exhibiting an explicit connection between these h*-polynomials and the non-nesting rook polynomials of Alexandersson and Jal (which are already known to be real-rooted), via an identity that preserves real-rootedness.

Significance. The result settles a recent conjecture in Ehrhart theory for a natural family of posets whose order polytopes are unimodularly equivalent to strength-one flow polytopes. The reduction to a known real-rooted family via a concrete generating-function identity supplies a direct combinatorial explanation and extends the reach of rook-theoretic techniques to width-two posets.

minor comments (3)
  1. [Abstract] Abstract, line 4: the parenthetical remark on P-Eulerian polynomials for width-two posets could be expanded by one sentence to clarify why the non-nesting rook polynomials are the appropriate intermediary.
  2. The identity relating the h*-polynomial to the non-nesting rook polynomial (the load-bearing step) should be stated as a numbered theorem or proposition with an explicit reference to the section where the real-rootedness transfer is verified.
  3. Ensure that all citations to Alexandersson-Jal appear with full bibliographic details in the reference list rather than the placeholder (2024+).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. We are pleased that the connection to non-nesting rook polynomials was viewed as providing a direct combinatorial explanation.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper proves an external conjecture (Lee et al.) by linking h*-polynomials of order polytopes to non-nesting rook polynomials via an identity that preserves real-rootedness. The rook polynomials originate in a separate prior publication (Alexandersson-Jal 2024+), and the present work does not reduce its central claim to a self-definition, fitted parameter renamed as prediction, or unverified self-citation chain. No equations or steps in the provided text exhibit a reduction of the target result to its own inputs by construction, satisfying the criteria for an independent mathematical argument.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The proof rests on standard Ehrhart theory, the cited equivalence of order polytopes to flow polytopes, and the real-rootedness of non-nesting rook polynomials from prior work; no free parameters, ad-hoc axioms, or new entities are introduced.

axioms (3)
  • standard math Standard properties of Ehrhart polynomials and the h*-transformation
    Invoked implicitly when discussing the h*-polynomials of the order polytopes.
  • domain assumption Unimodular equivalence between the order polytopes and strength-one flow polytopes for the dual graphs
    Cited from von Bell et al. (2022) as background.
  • domain assumption Non-nesting rook polynomials are real-rooted
    Cited from Alexandersson and Jal (2024+) and used to transfer the property.

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Reference graph

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