Stanley depth of weakly polymatroidal ideals and squarefree monomial ideals

Abstract

Let $I$ be a weakly polymatroidal ideal or a squarefree monomial ideal of a polynomial ring $S$. In this paper we provide a lower bound for the Stanley depth of $I$ and $S/I$. In particular we prove that if $I$ is a squarefree monomial ideal which is generated in a single degree, then $\mathrm{sdepth}(I)\ge n-\ell (I)+1$ and $\mathrm{sdepth}(S/I)\ge n-\ell (I)$, where $\ell (I)$ denotes the analytic spread of $I$. This proves a conjecture of the author in a special case.