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The Shareshian-Wachs conjecture: elementary positivity of chromatic quasisymmetric functions of natural unit interval graphs

For a weakly increasing hh with i≤h(i)≤ni\le h(i)\le n, the natural unit interval graph GG on [n][n] has edges {i,j}\{i,j\} with i<j≤h(i)i<j\le h(i). Shareshian and Wachs defined XG(x;q)=∑κqasc(κ)xκX_G(x;q)=\sum_\kappa q^{\mathrm{asc}(\kappa)}x_\kappa, summed over proper colorings and weighted by the number of edges i<ji<j with κ(i)<κ(j)\kappa(i)<\kappa(j), proved it symmetric and Schur-positive, and conjectured that its coefficients in the elementary basis lie in N[q]\mathbb N[q] (and are palindromic and unimodal). At q=1q=1 this is the Stanley-Stembridge conjecture via Guay-Paquet's reduction, proved by Hikita in 2024, whose formula gives positivity for each positive real qq but not coefficientwise. Does XG(x;q)X_G(x;q) expand in the elementary basis with coefficients in N[q]\mathbb N[q] for every natural unit interval graph?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Algebraic combinatorics; chromatic symmetric functions
Posed by
John Shareshian and Michelle L. Wachs
Year posed
2012
Years open
14y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
33 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: a terminating procedure assigns to each σ∈DG0\sigma\in D_G^0 (permutations whose adjacent descents are edges) a partition θG(σ)⊢n\theta_G(\sigma)\vdash n with XG(x;q)=∑σ∈DG0qginvG(σ)eθG(σ)(x)X_G(x;q)=\sum_{\sigma\in D_G^0}q^{\mathrm{ginv}_G(\sigma)}e_{\theta_G(\sigma)}(x), where ginvG\mathrm{ginv}_G counts inversions supported on edges. Hence every elementary coefficient lies in N[q]\mathbb N[q] and counts permutations; at q=1q=1 this reproves the Stanley-Stembridge conjecture (Hikita; Griffin-Mellit-Romero-Weigl-Wen). Not shown: the unimodality part of the Shareshian-Wachs conjecture, an efficient description of θG\theta_G (the procedure may be slow), or ee-positivity outside the natural unit interval class.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript, 'Elementary positivity of chromatic quasisymmetric functions' (September 24, 2026).

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture: it gives an explicit expansion XG=∑σ∈DG0qginvG(σ)eθG(σ)X_G=\sum_{\sigma\in D_G^0}q^{\mathrm{ginv}_G(\sigma)}e_{\theta_G(\sigma)}, which is the elementary-positivity part; the unimodality part is not claimed. Lean: lean/ComparatorChallenges/ElementaryPositivity.json exists (a definition challenge, OAI.elementaryPositivityWitness) and its solution module OAI.Combinatorics.Chromatic.Main is present at the pinned commit; the challenge is not in the formalization catalogue (formalization.yaml). The statement file was read here: for every natural unit interval graph given by a monotone extensive hh, the witness assigns a partition to each permitted nondescent permutation so that, for every number rr of variables, the chromatic polynomial over Polynomial N (proper colorings weighted by qascq^{\mathrm{asc}}) equals the sum of qginveθq^{\mathrm{ginv}}e_\theta. Holding for every finite rr gives the symmetric-function identity, so this states the headline. Not rebuilt here.

Sources

Changelog1 change

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