Revista de la
Unión Matemática Argentina
Depth and Stanley depth of powers of the path ideal of a cycle graph
Silviu Bălănescu and Mircea Cimpoeaş

Volume 68, no. 2 (2025), pp. 677–690    

Published online (final version): October 8, 2025

https://doi.org/10.33044/revuma.4641

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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 $S=K[x_1,\ldots,x_n]$. Let $d=\gcd(n,m)$. We prove that $\operatorname{depth}(S/J_{n,m}^t)\leq d-1$ for all $t\geq n-1$. We show that $\operatorname{sdepth}(S/J_{n,n-1}^t)=\operatorname{depth}(S/J_{n,n-1}^t)=\max\{n-t-1,0\}$ for all $t\geq 1$. Also, we give some bounds for $\operatorname{depth}(S/J_{n,m}^t)$ and $\operatorname{sdepth}(S/J_{n,m}^t)$, where $t\geq 1$.

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