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    The Partitionability Conjecture

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    Author's Accepted Manuscript (1.303Mb)
    Issue Date
    2017-02
    Author
    Duval, Art M.
    Klivans, Caroline J.
    Martin, Jeremy L.
    Publisher
    American Mathematical Society
    Type
    Article
    Article Version
    Scholarly/refereed, author accepted manuscript
    Published Version
    http://www.ams.org/journals/notices/201702/
    Rights
    This is the authors' accepted manuscript. The original publication is available at http://www.ams.org/journals/notices/201702/ .
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    Abstract
    In 1979 Richard Stanley made the following conjecture: Every Cohen-Macaulay simplicial complex is partitionable. Motivated by questions in the theory of face numbers of simplicial complexes, the Partitionability Conjecture sought to connect a purely combinatorial condition (partitionability) with an algebraic condition (Cohen-Macaulayness). The algebraic combinatorics community widely believed the conjecture to be true, especially in light of related stronger conjectures and weaker partial results. Nevertheless, in a 2016 paper [DGKM16], the three of us (Art, Carly, and Jeremy), together with Jeremy's graduate student Bennet Goeckner, constructed an explicit counterexample. Here we tell the story of the significance and motivation behind the Partitionability Conjecture and its resolution. The key mathematical ingredients include relative simplicial complexes, nonshellable balls, and a surprise appearance by the pigeonhole principle. More broadly, the narrative theme of modern algebraic combinatorics: to understand discrete structures through algebraic, geometric, and topological lenses.
    Description
    This is the authors' accepted manuscript. First published in Notices of the American Mathematical Society Volume 64 Issue 2, 2017, published by the American Mathematical Society.
    URI
    http://hdl.handle.net/1808/23760
    DOI
    https://doi.org/10.1090/noti1475
    Collections
    • Mathematics Scholarly Works [262]
    Citation
    Duval, A.M., Klivans, C.J., and Martin, J.L. The Partitionability Conjecture, Notices of the American Mathematical Society, 64:2 (2017), 117-122. http://dx.doi.org/10.1090/noti1475.

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    Lawrence, KS 66045
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    Contact KU ScholarWorks
    785-864-8983
    KU Libraries
    1425 Jayhawk Blvd
    Lawrence, KS 66045
    785-864-8983

    KU Libraries
    1425 Jayhawk Blvd
    Lawrence, KS 66045
    Image Credits
     

     

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