Stanley's partitionability conjecture for Cohen–Macaulay simplicial complexes

Let $\Delta$ be a simplicial complex. It is partitionable if its faces can be partitioned into intervals, each having a facet of $\Delta$ as its maximal element, with the interval's minimal element having the same dimension as the corresponding entry of the $h$-vector. A simplicial complex is Cohen–Macaulay over a field if its Stanley–Reisner ring has the Cohen–Macaulay property.

Stanley's partitionability conjecture. Every Cohen–Macaulay simplicial complex is partitionable.

Stanley described this as a central combinatorial conjecture on Cohen–Macaulay complexes. The conjecture is refuted by the explicit Cohen–Macaulay non-partitionable complex constructed in this paper.