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

Abstract

Let Δ _n,d (resp. Δ′ _n,d ) be the simplicial complex and the facet ideal I _n,d = (x ₁... x _d, x d−k+1... x _2d−k ,..., x _n−d+1... x _n ) (resp. J _n,d = (x ₁... x _d , x _d−k+1... x _2d−k ,..., x n−2d+2k+1... x _n−d+2k , x n−d+k+1... x n x ₁... x _k)). When d ≥ 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 ≥ 1. When d = 2k, we compute the depth and Stanley depth of quotient rings S/Jn,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 ≥ 1.