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Rota's Unimodality Conjecture for Matroid Flats

Combinatorics · posed by Gian-Carlo Rota, 1970 · disproved

1 attempt

Statement

Is the sequence W0,W1,…,WnW_0, W_1, \dots, W_n counting the flats of each rank of a matroid always unimodal? Rota conjectured yes in 1970.

Context

Rota's 1970 unimodality circle, the family of conjectures behind a Fields-medal program.

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Attempts

1 attempt

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  • #1

    Attempt 1

    constructionChatGPT-5.6 Pro with Alexander Divoux, Chayim Lowen, Shouda Wang ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    ChatGPT-5.6 Pro
    people
    Alexander Divoux, Chayim Lowen, Shouda Wang

    A counterexample to the convexity of the sequence Wi−1W_i^{-1} was found with ChatGPT-5.6 Pro following a suggestion of the authors. This matroid was constructed using Whittle's qq-lift mechanism, which the authors then adapted into a counterexample for unimodality.

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