arxiv: 2604.08394 · v2 · submitted 2026-04-09 · 🧮 math.CO

Ehrhart positivity for marked order polytopes

Katharina Jochemko , Krishna Menon This is my paper

Pith reviewed 2026-05-10 16:50 UTC · model grok-4.3

classification 🧮 math.CO

keywords Ehrhart positivitymarked order polytopesskew shapesposetsmultivariate polynomialsorder-preserving mapspiecewise polynomials

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The pith

A criterion on pairs of posets establishes that marked order polytopes of skew shapes are Ehrhart positive in the multivariate sense.

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

The paper examines pairs of finite posets A contained in P and the piecewise polynomial that counts the integer order-preserving extensions of any order-preserving map from A to the integers. It supplies a criterion that forces every coefficient in this multivariate polynomial to be nonnegative. The criterion holds for the poset pairs that define marked order polytopes of skew shapes, which therefore have Ehrhart polynomials with all coefficients nonnegative when tracked in multiple variables. The same criterion recovers the known positivity for ordinary order polytopes of skew shapes as a special case.

Core claim

Given a pair of finite posets A contained in P, the number of order-preserving extensions of a labeling on A is a piecewise polynomial in the values of that labeling. A combinatorial criterion on the pair ensures that all coefficients in this polynomial are nonnegative. Marked order polytopes of skew shapes satisfy the criterion, so their Ehrhart polynomials are nonnegative in the multivariate sense.

What carries the argument

The criterion for nonnegativity of coefficients in the multivariate piecewise polynomial that counts order-preserving extensions of maps from subposet A to poset P.

If this is right

Marked order polytopes of skew shapes have Ehrhart polynomials whose coefficients are all nonnegative.
The nonnegativity holds in the multivariate setting with one variable for each marked element.
The same criterion recovers the Ehrhart positivity already known for order polytopes of skew shapes.

Where Pith is reading between the lines

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

The criterion implies Ehrhart positivity for skew Gelfand-Tsetlin polytopes.
The criterion implies Ehrhart positivity for m-generalized Pitman-Stanley polytopes.
Any other poset pair that meets the combinatorial condition will likewise yield marked order polytopes with nonnegative Ehrhart coefficients.

Load-bearing premise

The combinatorial condition on the pair of posets A contained in P is sufficient to guarantee nonnegativity of every coefficient in the counting polynomial.

What would settle it

A pair of posets that satisfies the combinatorial condition but whose counting polynomial has a negative coefficient in some monomial would disprove the criterion.

Figures

Figures reproduced from arXiv: 2604.08394 by Katharina Jochemko, Krishna Menon.

**Figure 1.** Figure 1: The skew-shape 6533/211 and its associated poset. Their result implies Ehrhart positivity of the order polytope of such posets. A key ingredient of their proof is the following reduction to the nonnegativity of the linear term for families of posets that are closed under taking filters and ideals, together with a combinatorial formula for the linear term for skew-shaped posets [9, Proposition 4.1]. Theorem… view at source ↗

**Figure 2.** Figure 2: Marked poset associated to a generalized Pitman-Stanley polytope [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Marked poset associated to a skew Gelfand-Tsetlin polytope. We now show Ehrhart positivity for these polytopes using Proposition 2.1, thereby proving a conjecture of Alexandersson-Alhajjar [1, Conjecture 7] [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: Subposet S (in blue) containing J if ar = (4, 0) and ar+1 = (2, 6). When jr = 0 and jr+1 = m + 1, we consider the subposet (see [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

read the original abstract

Given a pair of finite posets $A \subseteq P$, the function counting integer-valued order preserving extensions of an order preserving map $\lambda : A\rightarrow \mathbb{Z}$ from $A$ to $P$ is given by a piecewise polynomial in $\lambda$. We provide a criterion for the nonnegativity of the coefficients of these multivariate polynomials and apply it to show that marked order polytopes of skew shapes are Ehrhart positive in a multivariate sense. This extends recent results of Ferroni-Morales-Panova on order polytopes of skew shapes and proves conjectures on the Ehrhart positivity of skew Gelfand-Tsetlin polytopes and $m$-generalized Pitman-Stanley polytopes due to Alexanderson-Alhajjar and Dugan-Hegarty-Morales-Raymond, respectively.

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.

Desk Editor's Note private letter to a colleague

Jochemko and Menon give a combinatorial criterion for nonnegative coefficients in the counting polynomials for order-preserving extensions and use it to prove multivariate Ehrhart positivity on skew marked order polytopes while settling two conjectures.

read the letter

The key takeaway is that Jochemko and Menon have found a combinatorial condition on a pair of posets A inside P that guarantees the piecewise polynomial counting order-preserving extensions has all nonnegative coefficients. They verify the condition for skew shapes and conclude that the associated marked order polytopes are Ehrhart positive in the multivariate sense. This both extends the Ferroni-Morales-Panova theorem on skew order polytopes and gives the first proofs of the Alexanderson-Alhajjar conjecture on skew Gelfand-Tsetlin polytopes and the Dugan-Hegarty-Morales-Raymond conjecture on m-generalized Pitman-Stanley polytopes. What stands out is the reduction of the positivity problem to a check on the poset pair rather than direct computation of the Ehrhart polynomial. The paper shows that the condition holds for the skew cases of interest, which is the main technical step. The identification of the counting function with the Ehrhart function of the marked order polytope is handled in a standard way and does not introduce new issues. The criterion itself is the genuinely new piece; it is not just a routine generalization of earlier work on order polytopes. The applications are clean because the condition turns out to be satisfied exactly when needed for the conjectures. A minor soft spot is that the paper focuses on the cases where the condition applies, so it leaves open how often the condition holds for arbitrary poset pairs. That is not a flaw in the argument, just a natural limit on the scope. There are no signs of circular reasoning or dependence on unverified assumptions in the reduction. This paper is aimed at combinatorialists working on Ehrhart theory, poset polytopes, and positivity questions in algebraic combinatorics. Anyone following the literature on Gelfand-Tsetlin or Pitman-Stanley polytopes will want to see the proofs. It is the kind of result that deserves a serious referee: it resolves open conjectures with a new general tool rather than ad-hoc arguments, and the logic appears sound. I would recommend sending it out for peer review.

Referee Report

0 major / 2 minor

Summary. The paper provides a criterion for the nonnegativity of coefficients in the multivariate piecewise polynomial counting integer-valued order-preserving extensions of an order-preserving map λ from a subposet A to P. It applies this criterion to prove that marked order polytopes of skew shapes are Ehrhart positive in the multivariate sense. This extends results of Ferroni-Morales-Panova on order polytopes of skew shapes and resolves conjectures on Ehrhart positivity for skew Gelfand-Tsetlin polytopes and m-generalized Pitman-Stanley polytopes.

Significance. If the criterion holds and its application to skew shapes is valid, the result is significant: it supplies a general combinatorial tool for establishing multivariate Ehrhart positivity for polytopes arising from pairs of posets A ⊆ P, unifies several positivity questions in the literature, and settles specific open conjectures. The reduction of the relevant polytopes to the marked-order setting is a strength, as is the explicit verification that the combinatorial condition holds for skew shapes.

minor comments (2)

[§1] The abstract refers to 'marked order polytopes' without a self-contained definition or pointer to the precise construction used here; adding a one-sentence reminder in §1 would improve accessibility for readers outside the immediate subfield.
Notation for the piecewise polynomial (e.g., the variables corresponding to the values of λ) is introduced in the statement of the criterion; a short table or diagram in the section presenting the criterion would clarify the correspondence between coordinates and poset elements.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, for highlighting its significance in unifying several positivity questions and resolving open conjectures, and for recommending minor revision. We will make the appropriate minor changes in the revised version.

Circularity Check

0 steps flagged

No significant circularity; self-contained combinatorial proof

full rationale

The paper defines a general criterion for nonnegativity of coefficients in the piecewise polynomial that counts order-preserving extensions of maps from A to P, then verifies that the criterion applies to the marked order polytopes of skew shapes by checking a stated combinatorial condition on the pair A ⊆ P. This verification is direct and independent of the target positivity statement; the multivariate Ehrhart function is identified with the counting function by standard definitions, and the argument extends prior results of Ferroni-Morales-Panova without reducing any load-bearing step to a self-citation, fitted parameter, or tautological renaming. No equations are shown to be equivalent by construction, and the central claim remains a verifiable combinatorial reduction rather than a circular re-expression of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard definitions of posets, order-preserving maps, and Ehrhart theory of polytopes; no free parameters or new entities are introduced.

axioms (2)

standard math Finite posets admit order-preserving extensions whose counting function is piecewise polynomial (standard Ehrhart theory for order polytopes).
Invoked in the first sentence of the abstract as background.
domain assumption Skew shapes induce marked order polytopes whose extension-counting polynomials satisfy the paper's nonnegativity criterion.
The application step assumes this combinatorial condition holds for skew shapes.

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discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

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