Stanley depth of weakly polymatroidal ideals and squarefree monomial ideals

TL;DR

This paper establishes lower bounds for the Stanley depth of weakly polymatroidal and squarefree monomial ideals, confirming a conjecture in a specific case and linking algebraic properties to combinatorial invariants.

Contribution

It provides new lower bounds for Stanley depth of certain monomial ideals and proves a conjecture for ideals generated in a single degree.

Findings

Lower bound for Stanley depth of weakly polymatroidal ideals.

Lower bound for Stanley depth of squarefree monomial ideals.

Confirmation of a conjecture in a special case.

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