On the Stanley length of monomial ideals

Let S=K[x₁,…,xₙ] be the ring of polynomials in n variables over an arbitrary field K. Given a finitely generated multigraded module M, its Stanley length, denoted by slength(M), is the minimal length of a Stanley decomposition of M. Let I⊂ S be a monomial ideal, minimally generated by m monomials. We give an upper bound for slength(I), in terms of its minimal monomial generators. Also, we give precise formulas for slength(I), if $n=2$ or $m=2$. Also, we show that if I has linear quotients, then slength(I)=m, and the converse holds in some special cases.