  The aim of this paper is to introduce a method for computing Hilbert
decompositions (and consequently the Hilbert depth) of a finitely generated
multigraded module $M$ over the polynomial ring $K[X_1,..., X_n]$ by reducing
the problem to the computation of the finite set of the new defined Hilbert
partitions. Moreover, in the last section, we show that Hilbert partitions may
also be used for computing the Stanley depth of the module $M$.
