On the Stanley depth of squarefree monomial ideals

TL;DR

This paper proves Stanley's conjecture for certain squarefree monomial ideals associated with chordal clutters and introduces the Schmitt--Vogel number, establishing bounds on Stanley depth.

Contribution

It establishes Stanley's conjecture for edge ideals of d-complements of chordal clutters and introduces the Schmitt--Vogel number with new depth bounds.

Findings

Stanley's conjecture holds for edge ideals of d-complements of chordal clutters.

Introduces Schmitt--Vogel number and relates it to Stanley depth.

Provides inequalities linking Schmitt--Vogel number and Stanley depth.

Abstract

Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over the field $\mathbb{K}$. Suppose that $\mathcal{C}$ is a chordal clutter with $n$ vertices and assume that the minimum edge cardinality of $\mathcal{C}$ is at least $d$. It is shown that $S/I(c_d(\mathcal{C}))$ satisfies Stanley's conjecture, where $I(c_d(\mathcal{C}))$ is the edge ideal of the $d$-complement of $\mathcal{C}$. This, in particular shows that $S/I$ satisfies Stanley's conjecture, where $I$ is a quadratic monomial ideal with linear resolution. We also define the notion of Schmitt--Vogel number of a monomial ideal $I$, denoted by ${\rm sv}(I)$ and prove that for every squarefree monomial ideal $I$, the inequalities ${\rm sdepth}(I)\geq n-{\rm sv}(I)+1$ and ${\rm sdepth}(S/I)\geq n-{\rm sv}(I)$ hold.