Stanley depths of certain Stanley-Reisner rings

Let Δ be a simplicial complex on [n], S=K[x1,…,xn] the polynomial ring in n-variables over a field K and K[Δ]=S/IΔ the Stanley-Reisner ring of Δ with respect to K. It is proved that the Stanley Conjecture holds for K[Δ], i.e., sdepthS(K[Δ])≥depthS(K[Δ]), when IΔ has four associated prime ideals. It is also obtained that sdepthS(K[Δ])≥sizeS(IΔ).