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

Abstract

Let $I_{n,m}:=(x_1{x}_2\cdots x_m,x_2{x}_3\cdots x_{m+1},\dots ,x_{n-m+1}\cdots x_n)$ be the $m$-path ideal of the path graph of length $n$, in the ring $S=K[x_1,\dots ,x_n]$. We prove that: $$\mathtt{depth}(S/I_{n,m}^t)=\{\begin{array}{ll}n-t+2-\lfloor \frac{n-t+2}{m+1}\rfloor -\lceil \frac{n-t+2}{m+1}\rceil ,& t\le n+1-m\\ m-1,& t>n+1-m\end{array},\text{ for all }t\ge 1.$$ Also, we prove that $\mathtt{depth}(S/I_{n,m})\ge \mathtt{sdepth}(S/I_{n,m}^t)\ge \mathtt{depth}(S/I_{n,m}^t)$ and $\mathtt{sdepth}(I_{n,m}^t)\ge \mathtt{depth}(I_{n,m}^t)$, for all $t\ge 1$.

16 pages