# 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.

## Key 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.