## “School of Mathematics”

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Paper   IPM / M / 13736
 School of Mathematics Title: Two lower bounds for the Stanley depth of monomial ideals Author(s): S. A. Seyed Fakhari (Joint with L. Katthan) Status: To Appear Journal: Math. Nachr. Supported by: IPM
Abstract:
Let J\varsubsetneq I be two monomial ideals of the polynomial ring S=\mathbbK[x1,…,xn]. 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) ≥ nl(I/J)+1 holds. On the other hand, we show that \sdepth(I/J) ≥ n−dimLI/J, where dimLI/J denotes 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.