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 $\operatorname{sdepth} (I)\geq n-\ell(I)+1$ and $\operatorname{sdepth} (S/I)\geq n-\ell(I)$, where $\ell(I)$ denotes the analytic spread of $I$. This proves a conjecture of the author in a special case.