-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 -vertex example refutes it: for the Hessenberg function and the coefficients of are , and . 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
- —
- Years open
- —
- 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 arXiv preprint, not yet peer-reviewed.