Depth and Stanley depth of powers of the path ideal of a path graph

TL;DR

This paper investigates the depth and Stanley depth of powers of the path ideal of a path graph, providing explicit formulas and inequalities that relate these invariants.

Contribution

It derives exact formulas for the depth of powers of the path ideal and establishes inequalities connecting depth and Stanley depth.

Findings

Explicit formula for depth of $S/I_{n,m}^t$ for all $t",

Inequalities relating Stanley depth and depth for these ideals

Abstract

Let $I_{n,m}:=(x_1x_2\cdots x_m,\; x_2x_3\cdots x_{m+1},\; \ldots,\; x_{n-m+1}\cdots x_n)$ be the $m$-path ideal of the path graph of length $n$, in the ring $S=K[x_1,\ldots,x_n]$. We prove that: $$\mathtt{depth}(S/I_{n,m}^t)=\begin{cases} n-t+2 - \left\lfloor \frac{n-t+2}{m+1} \right\rfloor - \left\lceil \frac{n-t+2}{m+1} \right\rceil, & t \leq n+1-m \\ m-1,& t > n+1-m \end{cases},\text{ for all }t\geq 1.$$ Also, we prove that $\mathtt{depth}(S/I_{n,m}) \geq \mathtt{sdepth}(S/I_{n,m}^t) \geq \mathtt{depth}(S/I_{n,m}^t)$ and $\mathtt{sdepth}(I_{n,m}^t)\geq \mathtt{depth}(I_{n,m}^t)$, for all $t\geq 1$.