Stanley depth of weakly 0-decomposable ideals

Recently, Seyed Fakhari proved that if I is a weakly polymatroidal monomial ideal in \({S\,=\,\mathbb{K}[x_1,\ldots,x_n]}\), then Stanley’s conjecture holds for S/I, namely, sdepth \({(S/I)\,\geq\, {\rm depth}(S/I)}\). We generalize his ideas and introduce several new classes of monomial ideals which also share this property. In particular, if I is the Stanley–Reisner ideal of the Alexander dual of a nonpure vertex decomposable simplicial complex, then Stanley’s conjecture holds for S/I.