Two lower bounds for the Stanley depth of monomial ideals

Let $J⊊I$ be two monomial ideals of the polynomial ring $S=\mathbb{K}[x_1,\dots ,x_n]$. In this paper, we provide two lower bounds for the Stanley depth of $I\mathrm{/}J$. On the one hand, we introduce the notion of lcm number of $I\mathrm{/}J$, denoted by $l(I\mathrm{/}J)$, and prove that the inequality ${\rm sdepth}(I/J)\geq n-l(I/J)+1$hold\mathrm{.}Ontheotherhand,weshowthat${\sdepth}(I/J)\geq n-\dim L_{I/J}$,where$\dim L_{I/J}$ denotes the order dimension of the lcm lattice of $I\mathrm{/}J$. We show that $I$ and $S\mathrm{/}I$ satisfy Stanley’s conjecture, if either the lcm number of $I$ or the order dimension of the lcm lattice of $I$ is small enough. Among other results, we also prove that the Stanley–Reisner ideal of a vertex decomposable simplicial complex satisfies Stanley’s conjecture.