Combinatorial Reductions for the Stanley Depth of $I$ and $S/I$

TL;DR

This paper introduces combinatorial methods to compare the Stanley depth of monomial ideals and their quotients, proving new inequalities for small polynomial rings and extending results via computational search.

Contribution

It develops combinatorial tools to relate Stanley depths of ideals and their quotients, providing new bounds for small polynomial rings and partial answers to existing conjectures.

Findings

Stanley depth of S/I is larger than I for rings with up to 5 indeterminates.

Extended inequality holds for rings with up to 7 indeterminates using computer search.

Partially answers questions posed by Propescu, Qureshi, and Herzog.

Abstract

We develop combinatorial tools to study the relationship between the Stanley depth of a monomial ideal $I$ and the Stanley depth of its compliment, $S/I$. Using these results we are able to prove that if $S$ is a polynomial ring with at most 5 indeterminates and $I$ is a square-free monomial ideal, then the Stanley depth of $S/I$ is strictly larger than the Stanley depth of $I$. Using a computer search, we are able to extend this strict inequality up to polynomial rings with at most 7 indeterminates. This partially answers questions asked by Propescu and Qureshi as well as Herzog.