Let $K$ be a field and $S=K[x_1,...,x_n]$ a the polynomial ring in $n$ variable over the field $K$ and $M$ a finitely generated $\zz^n$-graded $S$ module. In this talk we recal some recentresults considering Stanley conjecture on Stanley decomposition. We also study $h4-regularity conjecture on Stanley decomposition. We show thatStanley conjecturewhich sayes that any Cohen-Macaulay simplicial complex is paetitionable is a special case of Staleyconjecture on Stanley decomposition. It is also shown that Stanley depth of quotient of monomial ideals can be computed in finitely many step
