Geodesic
A non-partitionable Cohen–Macaulay simplicial complex
Discrete mathematics & theoretical computer science, DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016), DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016) (2020).

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A long-standing conjecture of Stanley states that every Cohen–Macaulay simplicial complex is partition- able. We disprove the conjecture by constructing an explicit counterexample. Due to a result of Herzog, Jahan and Yassemi, our construction also disproves the conjecture that the Stanley depth of a monomial ideal is always at least its depth. 
@article{DMTCS_2020_special_379_a7,
     author = {Duval, Art M. and Goeckner, Bennet and Klivans, Caroline J. and Martin, Jeremy},
     title = {A non-partitionable {Cohen{\textendash}Macaulay} simplicial complex},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)},
     year = {2020},
     doi = {10.46298/dmtcs.6325},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.6325/}
}
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Duval, Art M.; Goeckner, Bennet; Klivans, Caroline J.; Martin, Jeremy. A non-partitionable Cohen–Macaulay simplicial complex. Discrete mathematics & theoretical computer science, DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016), DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016) (2020). doi : 10.46298/dmtcs.6325. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.6325/

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