Two lower bounds for the Stanley depth of monomial ideals

Let J ⊈ I be two monomial ideals of the polynomial ring S = K [ x 1 , ... , x n ] . In this paper, we provide two lower bounds for the Stanley depth of I / J . On the one hand, we introduce the notion of lcm number of I / J , denoted by l ( I / J ) , and prove that the inequality sdepth ( I / J ) ≥ n − l ( I / J ) + 1 holds. On the other hand, we show that sdepth ( I / J ) ≥ n − dim L I / J, where dim L I / Jdenotes the order dimension of the lcm lattice of I / J . We show that I and S / 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.