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The Partitionability Conjecture
Duval, Art M. Klivans, Caroline J. Martin, Jeremy L.
Duval, Art M.
Klivans, Caroline J.
Martin, Jeremy L.
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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.
Date
2017-02
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American Mathematical Society
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Keywords
Partitionability conjecture, Combinatorics
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.