On the Stanley length of monomial ideals

Let $S=K[x_1,\dots ,x_n]$ 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 $\mathrm{slength}(M)$, is the minimal length of a Stanley decomposition of $M$. Let $I\subset S$ be a monomial ideal, minimally generated by $m$ monomials. We give an upper bound for $\mathrm{slength}(I)$, in terms of its minimal monomial generators. Also, we give precise formulas for $\mathrm{slength}(I)$, if $n=2$ or $m=2$. Also, we show that if $I$ has linear quotients, then $\mathrm{slength}(I)=m$, and the converse holds in some special cases.