# On the Stanley depth of powers of monomial ideals

## TL;DR

This paper surveys recent research on the Stanley depth of powers, integral closures, and symbolic powers of monomial ideals, exploring cases where Stanley's conjecture holds or fails.

## Contribution

It compiles recent findings on the behavior of Stanley depth for various powers of monomial ideals, highlighting conditions where the conjecture is valid or invalid.

## Key findings

- Stanley's conjecture is disproved in general but may hold for high powers.
- Recent results show the Stanley depth of powers can stabilize or follow specific patterns.
- The survey summarizes conditions under which Stanley's inequality is true for monomial ideals.

## Abstract

Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over $\mathbb{K}$. In 1982, R. Stanley associated a combinatorial invariant to any finitely generated $\mathbb{Z}^n$-graded $S$-module which is now called Stanley depth. Stanley conjectured that this invariant is an upper bound for the depth of module. Stanley's conjecture has been disproved by Duval et al. \cite{abcj}, and the counterexample is a quotient of squarefree monomial ideals. On the other hand, there are evidences showing that Stanley's inequality can be true for high powers of monomial ideals. In this survey article, we collect the recent results in this direction. More precisely, we investigate the Stanley depth of powers, integral closure of powers and symbolic powers of monomial ideals.