The Partitionability Conjecture

View/ Open
Issue Date
2017-02Author
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/ .
Metadata
Show full item recordAbstract
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.
Collections
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.
Items in KU ScholarWorks are protected by copyright, with all rights reserved, unless otherwise indicated.
We want to hear from you! Please share your stories about how Open Access to this item benefits YOU.