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ee-Log-Concavity of Chromatic Quasisymmetric Functions

Chromatic quasisymmetric functions of natural unit interval graphs were conjectured to have log-concave coefficients in the elementary basis. A connected 1313-vertex example refutes it: for the Hessenberg function h=(2,4,4,6,7,10,10,10,10,12,12,13,13)h=(2,4,4,6,7,10,10,10,10,12,12,13,13) and λ=(6,5,1,1)\lambda=(6,5,1,1) the coefficients of q5,q6,q7q^5,q^6,q^7 are 1,6,381,6,38, and 62<1⋅386^2 < 1 \cdot 38. The coefficient is still positive, palindromic and unimodal, so log-concavity is separated from the weaker shape properties that motivated the conjecture.

Result
Disproved
Status
Resolved
AI contribution
AI-assisted
Method
Computation
Field
Algebraic combinatorics
Posed by
Bruce Sagan, Foster Tom
Year posed
2026
Years open
0y
Solved
2026-07-22
Model
ChatGPT with Codex
Vendor
OpenAI
Collaborators
Boris Kafidov
Verification
Site-confirmed
Publication
Preprint
Significance
15 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What the AI did

The disclosure describes a computer-assisted investigation in which the model was used extensively as an interactive research assistant: it generated and revised the exploratory and verification code, ran independent computational cross-checks, searched the literature, and helped organize and draft the manuscript. The author states that no assertion is accepted on the authority of model output.

Verification

The counterexample is a single explicit graph and partition, and the paper ships an exact standard-library Python verifier with a recorded SHA-256 digest. arXiv preprint, not yet peer-reviewed.

Sources

Changelog3 changes
  • Matarisvanset Year posed to “2026”, also Significance note, Significance note, More links
  • Rasmus Lindahlchanged Name from “e-Log-Concavity of Chromatic Quasisymmetric Functions” to “$e$-Log-Concavity of Chromatic Quasisymmetric Functions”
  • Rasmus Lindahlchanged Name from “$e$-Log-Concavity of Chromatic Quasisymmetric Functions” to “e-Log-Concavity of Chromatic Quasisymmetric Functions”

Discussion