Depth and Stanley depth of powers of the path ideal of a cycle graph. II

Abstract

Let $J_{n,m}:=(x_1x_2\cdots x_m,\; x_2x_3\cdots x_{m+1},\; \ldots,\; x_{n-m+1}\cdots x_n,\; x_{n-m+2}\cdots x_nx_1, \ldots, x_nx_1\cdots x_{m-1})$ be the $m$-path ideal of the cycle graph of length $n$, in the ring of polynomials $S=K[x_1,\ldots,x_n]$. As a continuation of arxiv:2303.15032v2, we prove several new results regarding $\operatorname{depth}(S/J_{n,m}^t)$ and $\operatorname{sdepth}(S/J_{n,m}^t)$, where $t\geq 1$.

Theorems & Definitions (31)

• Lemma 1.1
• Lemma 1.2
• Lemma 1.3
• Lemma 1.4
• Lemma 1.5
• Lemma 1.6
• Lemma 1.7
• Theorem 1.8
• Theorem 1.9
• Lemma 1.10