# Depth and Stanley depth of the facet ideals of some classes of simplicial complexes

• Volume: 67, Issue: 3, page 753-766
• ISSN: 0011-4642

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## Abstract

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Let ${\Delta }_{n,d}$ (resp. ${\Delta }_{n,d}^{\text{'}}$) be the simplicial complex and the facet ideal ${I}_{n,d}=\left({x}_{1}\cdots {x}_{d},{x}_{d-k+1}\cdots {x}_{2d-k},...,{x}_{n-d+1}\cdots {x}_{n}\right)$ (resp. ${J}_{n,d}=\left({x}_{1}\cdots {x}_{d},{x}_{d-k+1}\cdots {x}_{2d-k},...,{x}_{n-2d+2k+1}\cdots {x}_{n-d+2k},{x}_{n-d+k+1}\cdots {x}_{n}{x}_{1}\cdots {x}_{k}\right)$). When $d\ge 2k+1$, we give the exact formulas to compute the depth and Stanley depth of quotient rings $S/{J}_{n,d}$ and $S/{I}_{n,d}^{t}$ for all $t\ge 1$. When $d=2k$, we compute the depth and Stanley depth of quotient rings $S/{J}_{n,d}$ and $S/{I}_{n,d}$, and give lower bounds for the depth and Stanley depth of quotient rings $S/{I}_{n,d}^{t}$ for all $t\ge 1$.

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