A non-partitionable Cohen-Macaulay simplicial complex

TL;DR

This paper constructs a counterexample to Stanley's conjecture that all Cohen-Macaulay simplicial complexes are partitionable, also disproving related conjectures about Stanley depth and ideal depth.

Contribution

It provides the first explicit counterexample to the long-standing conjecture linking Cohen-Macaulay complexes and partitionability.

Findings

Counterexample disproves the conjecture

Disproves Stanley depth conjecture for monomial ideals

Challenges previous assumptions in combinatorial commutative algebra

Abstract

A long-standing conjecture of Stanley states that every Cohen-Macaulay simplicial complex is partitionable. 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.