Betti posets and the Stanley depth

Let $S$ be a polynomial ring and let $I \subseteq S$ be a monomial ideal. In this short note, we propose the conjecture that the Betti poset of $I$ determines the Stanley projective dimension of $S/I$ or $I$. Our main result is that this conjecture implies the Stanley conjecture for $I$, and it also implies that $\operatorname{sdepth} S/I \geq \operatorname{depth} S/I - 1.$ Recently, Duval et al. found a counterexample to the Stanley conjecture, and their counterexample satisfies $\operatorname{sdepth} S/I = \operatorname{depth} S/I - 1$. So if our conjecture is true, then the conclusion is best possible.

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