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Exact thresholds for Schur positivity of the lattices $\mathbf m\times\mathbf 2$ and $\mathbf m\times\mathbf 3$

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Pith's one-line read The paper determines the exact sizes at which the chain-product lattices m×2 and m×3 stop being Schur positive, exhibiting explicit negative Schur coefficients for all m beyond the threshold and settling the remaining small cases by exact c

arxiv 2510.03116 v2 pith:26SS73X5 submitted 2025-10-03 math.CO

David G.L. Wang , K. Zhang This is my paper
classification math.CO MSC 05E0505A1506A07
keywords chromaticsymmetricfunctionSchurpositivityspecialribbontabloidstablecompositionPieri'sruleproductofchainsincomparabilitygraphstronglyniceproperty
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that the chain-product lattices m×2 and m×3 stop being Schur positive at precise sizes: m×2 is Schur positive exactly for m≤7, and m×3 exactly for m≤6. For every m≥8 the authors exhibit an explicit Schur coefficient that is negative, computed via Pieri's rule and a combinatorial formula for Schur coefficients in terms of special ribbon tabloids. The m=7 case of m×3, which falls below the infinite family, is settled by exact computer computation and also has a negative coefficient. These results confirm, for n=2 and n=3, a conjecture about when products of chains fail to be Schur positive, and they show that m×3 is not even strongly nice for m≥44.

What carries the argument

The main tool is the special ribbon tabloid formula, which expresses any Schur coefficient of a chromatic symmetric function as a signed sum of stable-composition counts over shapes obtained by decomposing the Young diagram into ribbons with heads in the first column. Pieri's rules convert the s_1^2 factor from the two isolated vertices into a small set of partitions that can be examined individually. The structural lemma that every stable set in H_m^n is 'southwest-to-northeast' (an anti-blocking order) reduces the stable-composition counts to binomial sums that are evaluated explicitly.

What would settle it

Run an exact computer-algebra computation of the full Schur expansion of the chromatic symmetric function of the incomparability graph of the 7×3 lattice. The paper claims a negative Schur coefficient; if all coefficients are nonnegative, the exact threshold m≤6 is wrong. A second check: for the 7×2 lattice, the paper claims all Schur coefficients are nonnegative; finding one negative would falsify the m≤7 threshold.

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Extended reading notes

Core claim

The central discovery is that Schur positivity of the incomparability graphs of m×n chain products has exact finite thresholds in the first two non-trivial width cases. For m×2, the coefficient of s_{(m−2)2^4} in X_{inc(m×2)} equals −m(m−2)(m−7)/3 for m≥8, negative precisely when m≥8; combined with verified positivity for m≤7 this gives the exact threshold. For m×3, the coefficient of s_{(m+2)(m−3)^2 4} is −4/3 m^3 + 16m^2 − 176/3 m + 96 for m≥8, negative for m≥8, and the finite cases through m=6 are positive while m=7 has a negative coefficient, yielding the exact threshold m≤6. The proof uses the fact that the two extremal elements of the lattice become isolated vertices, reducing X_inc(m×

Load-bearing premise

The exact-threshold claims rest on the correctness and completeness of exact computer computations for m≤7 in m×2 and m≤6 plus m=7 in m×3, for which the paper supplies no code or output; an error in any of these finite checks would change the thresholds even though the m≥8 results remain valid.

Editorial extensions

If this is right

  • m×2 is Schur positive for m≤7 and not for m≥8; m×3 is Schur positive for m≤6 and not for m≥7, so both thresholds are now known exactly.
  • The explicit negative Schur coefficients for all m≥8 provide a deterministic certificate of failure that does not require computer verification.
  • The confirmed n=2 and n=3 cases support the general conjecture that m×n is not Schur positive for m≥n+5 (with m≥8 when n=2).
  • For m≥44, m×3 is not strongly nice, so even the weakly monotone stable-composition property fails for these distributive lattices.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same Pieri-based approach may extend to n=4, but the number of ribbon tabloids and stable-composition cases will grow rapidly; the existence of a uniform negative coefficient for all m in a range suggests a pattern that might be proven by induction.
  • The exact thresholds (7 for m×2, 6 for m×3) hint that for fixed n the threshold might be roughly n+4 or n+5, matching the conjecture's m≥n+5.
  • The strong-niceness failure at m≥44 is likely not sharp; the combinatorial formulas could be used to search for the true strong-niceness threshold, which may be much lower.
  • If the finite computer checks were wrong, the infinite-family results still stand, but the exact-threshold statements would need revision; providing scripts would make the finite cases independently checkable.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; negative coefficients are computed from explicit stable-composition counts via a general published formula.

full rationale

Walking the derivation chain: the negative Schur coefficients are obtained by applying Theorem 2.1 (Wang–Wang formula) to explicit stable-composition counts (Propositions 3.2, 4.1–4.6), then passing through Pieri's rules to account for the two isolated vertices (Theorems 3.4 and 4.8). Each input is a general parameter-free counting formula or a composition count; none of the inputs is defined in terms of, or fitted to, the target coefficient. The one self-citation, Theorem 2.1 from [15], is load-bearing but independent: it is a published formula for arbitrary graphs with stated assumptions and no dependence on Schur positivity of m×n. The finite thresholds (m≤7, m≤6) rest on exact SageMath computations not reproduced in the paper; this is a reproducibility/correctness concern, not circularity, since those computations are external verification of finite cases rather than a renamed form of the claim. I find no step where Eq. X reduces to Eq. Y by construction, no fitted parameter called a prediction, and no uniqueness theorem or ansatz smuggled via self-citation.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No fitted numerical parameters and no new postulated entities. The exhibited partitions are explicit witnesses, not tuned constants. The main load-bearing inputs are the Wang–Wang formula, Pieri's rules, and the unshipped SageMath computation for finite cases.

assumptions (3)
  • standard math Theorem 2.1 (Wang–Wang formula) correctly expresses any Schur coefficient as a signed sum over special ribbon tabloids of stable-composition counts.
    Published general combinatorial formula (Discrete Appl. Math. 2020) by one of the present authors and another author. It is parameter-free, applies to all graphs, and does not assume the target result, so it has independent grounding.
  • standard math Pieri's rules (Proposition 2.3) correctly describe multiplication by s_1^2 = s_2+s_{11}.
    Classical result in symmetric function theory, used to handle the two isolated vertices of inc(m×n).
  • ad hoc to paper The exact SageMath computations settling the finite cases (m×2 for m≤7, m×3 for m≤6, and negativity of 7×3) are correct and complete.
    The abstract states the finite cases are settled by exact SageMath computations, but no code, scripts, or output are provided. The 'exact threshold' claim depends on this unverified computational assertion.

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Pith. "Pith review of Exact thresholds for Schur positivity of the lattices $\mathbf m\times\mathbf 2$ and $\mathbf m\times\mathbf 3$." pith.science (2026). https://pith.science/paper/26SS73X5

@misc{pith2026251003116,
  author       = {Pith},
  title        = {Pith review of: Exact thresholds for Schur positivity of the lattices $\mathbf m\times\mathbf 2$ and $\mathbf m\times\mathbf 3$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/26SS73X5}},
  note         = {Machine review of arXiv:2510.03116}
}
abstract

We determine the exact thresholds for Schur positivity in the two families of chain products $\mathbf m\times\mathbf2$ and $\mathbf m\times\mathbf3$: the former is Schur positive exactly for $m\le7$, and the latter exactly for $m\le6$. For $m\ge8$, we prove non-Schur-positivity in both families by exhibiting explicit negative Schur coefficients obtained from Pieri's rules and stable-composition counts. The remaining finite cases are settled by exact SageMath computations; in particular, $\mathbf7\times\mathbf3$ has a negative Schur coefficient. These results settle the $n=2$ and $n=3$ cases in the conjectural picture of Li, Qiu, Yang, and Zhang and sharpen the $n=3$ boundary by one. We also show that $\mathbf m\times\mathbf3$ is not strongly nice for $m\ge44$.

Figures

Figures reproduced from arXiv: 2510.03116 by the authors.

Figure 1
Figure 1. The lattice m × 2 and the graph Gm. j > i. As noted by Erd˝os [4], “the name half-graph comes from the fact that it can be considered to be half of a complete bipartite graph.” We begin with a basic counting observation that will serve as a key input. Lemma 3.1. For any 0 ≤ a, b, c ≤ m with a + b + c = 2m, sc12(Gm; abc) =     m − 1 c  , if a = m or b = m,  m c  , if a, b ≤ m − 1. Proof. Let (A, B, C) ∈ SC… view at source ↗
Figure 2
Figure 2. The set T(m−2)24 = {κ1, . . . , κ6}. = −2  1 +  m 2  + 2 1 +  m 3  + 2 2m +  m 2  − 2  m +  m 2  +  m 3  − 2  m +  m 3  +  m 4  + 2 2  m 2  +  m 4  = − m(m − 1)(m − 5) 3 , which is negative for m ≥ 6. This shows that Gm is not Schur positive. In the same way, we deduce [s(m−1)(m−2)3]XGm = Nm(m−1)1 − N(m−1)22 − Nm(m−3)3 + N(m−1)(m−2)3 = 0, [s(m−1)(m−3)4]XGm = Nm(m−2)2 − N(m−1)(m−2)3 − Nm… view at source ↗
Figure 3
Figure 3. The set {ν1, . . . , ν4} of partitions such that [sλ]sνi s 2 1 ̸= 0. The shaded circles indicate the removed cells. removing operations, since the shaded circles for ν2 form both a horizontal strip and a vertical strip. Let H = Gm−1. By Propositions 2.3 and 3.3, we then compute [sλ]Xinc(m×2) = [sλ]s 2 1XH = [sν1 ]XH + 2[sν2 ]XH + [sν3 ]XH + [sν4 ]XH = −(m − 1)(m − 2)(m − 6)/3 + 2 · 0 + 0 + (2m − 4) = −m(m − 2)(m − 7… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: The lattice m × 3 and the graph H3 m. letting vk2 = xk, vk1 = yk and vk0 = zk for all k, and denote the relabeled graph by Hm. The graph has order 3m − 2. More precisely, let X = {x0, . . . , xm−2}, Y = {y0, . . . , ym−1} and Z = {z1, . . . , zm−1}. Then Hm has vertex …
Figure 5
Figure 5. Figure 5: The decompositions of X, Y and Z. In fact, Eq. (4.8) allows us to locate the subsets A ∩ X, A ∩ Y and A ∩ Z. By Lemma 2.4 and by the premise (x0, zm−1) ∈ B × C, we deduce that B ⊆ {y0} ∪ X0(j−1), C ⊆ Z(k+1)(m−1) ∪ {ym−1}, and Y1(k−1) ∪ Y(j+1)(m−2) ⊆ D. We claim that X0…
Figure 6
Figure 6. Figure 6: The set Tm(m−3)24 = {κ1, . . . , κ6}. as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

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