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e-Log-Concavity of Chromatic Quasisymmetric Functions

Combinatorics · posed by Bruce Sagan, Foster Tom · disproved

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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.

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. computation · #1

    Boris Kafidov, using ChatGPT with Codex

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

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    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.

  2. Recorded elsewhere on #1 · not checked here

    recorded: correctVibeMathed 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 arXiv preprint, not yet peer-reviewed.

    Repeated from the source; nothing was checked here.

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