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$e$-Log-Concavity of Chromatic Quasisymmetric Functions

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e-log-concavity-chromatic-quasisymmetricCombinatoricsposed by Bruce Sagan, Foster Tomrecorded: disproved

1 attempt · 1 machine check · no person has looked

Statement

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.

Context

A conjecture in the Shareshian-Wachs chromatic quasisymmetric function programme, verified computationally by Abreu and Nigro up to the order just below this witness.

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  • #1

    Attempt 1

    computationChatGPT with Codex with Boris Kafidov ·
    AI involvement
    ai assisted
    — a person led the work and used a model along the way.
    models
    ChatGPT with Codex
    people
    Boris Kafidov

    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.

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    0 human reviews · 1 machine check

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    • Machine check · not human verification

      machine: correct

      Recorded from VibeMathed site check ·

      scope Reproduction by the VibeMathed site

      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 arXiv preprint, not yet peer-reviewed.

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