Shareshian–Wachs elementary unimodality for natural unit interval graphs

For a weakly increasing $h:[n]\to[n]$ with $i\le h(i)$ , the natural unit interval graph $G$ on $[n]$ has edges $\{i,j\}$ with $i<j\le h(i)$ . Its chromatic quasisymmetric function is $X_G(x;q)=\sum_\kappa q^{\mathrm{asc}(\kappa)}x_\kappa$ , summed over proper colorings, where $\mathrm{asc}(\kappa)$ counts edges $i<j$ with $\kappa(i)<\kappa(j)$ ; Shareshian and Wachs proved it symmetric and palindromic in $q$ . Write $X_G=\sum_{j=0}^{m}a_j(x)q^j$ with $m=|E(G)|$ . Conjecture 1.3 asserts that each $a_j$ is $e$ -positive and that $a_{j+1}-a_j$ is $e$ -positive for $0\le j<(m-1)/2$ ; equivalently, every coefficient $c_{G,\lambda}(q)$ of $e_\lambda$ is a nonnegative, palindromic, unimodal polynomial. Is this true for every natural unit interval graph?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-assisted
Method: Argument
Field: Algebraic combinatorics; chromatic symmetric functions
Posed by: John Shareshian and Michelle L. Wachs, Chromatic quasisymmetric functions, Conjecture 1.3 (arXiv:1405.4629; Adv. Math. 295 (2016), 497-551)
Year posed: —
Years open: —
Solved: 2026-10-22
Model: GPT 6 Pro; Claude Fable 5.1
Vendor: OpenAI; Anthropic
Collaborators: Mingchang Liu
Verification: Unreviewed
Publication: Preprint
Significance: 26 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for every natural unit interval graph $G$ with $e$ edges and every $\lambda\vdash|V(G)|$ , $c_{G,\lambda}(q)=\sum_{j=0}^{\lfloor e/2\rfloor}a_{G,\lambda,j}q^j(1+q+\cdots+q^{e-2j})$ with $a_{G,\lambda,j}\in\mathbb N$ , so every elementary coefficient is nonnegative, palindromic about $e/2$ and unimodal. With positivity this is the whole of Conjecture 1.3. The proof builds rigid quiver potentials with nilpotent Jacobian algebras and uses purity and hard Lefschetz for BPS sheaves. It inherits the triangular charts, rotations and section formalism of OpenAI's unrefereed September 2026 positivity manuscript (its Sections 2 and 4-5, not its positivity proof in Section 3), so it depends on that framework. Log-concavity, which is stronger, fails in general.

What the AI did

The manuscript reports collaborations between the author and AI models. GPT 6 Pro and Claude Fable 5.1 assisted with mathematical exploration, proof development and checking, and manuscript drafting and revision. The author takes responsibility for the mathematical results and manuscript contents.

Verification

No independent mathematician has checked this yet. Read by this site on 3 November 2026. The Zenodo record (v1, 22 October 2026, sole author Mingchang Liu, 31 pages) was opened; Theorem 1.1 was read against Conjecture 1.3 in the arXiv version of Shareshian-Wachs and states it in full, given the known palindromicity. The AI statement is in the manuscript. A separate computation written here expanded $X_G$ in the elementary basis for all 132 natural unit interval graphs on up to 6 vertices (1,836 coefficients) and found every coefficient nonnegative, palindromic and unimodal; it says nothing about larger graphs. The proof was not read, and its dependence on the unrefereed OpenAI framework was not assessed.

Sources

Submitted by SilentIbis759 on 22 Oct 2026

Shareshian–Wachs elementary unimodality for natural unit interval graphs

What was actually shown

What the AI did

Verification

Sources

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