It is proved that if an algebra R over a field can be endowed with a pointed and finite-dimensional ℕn-filtration such that the associated ℕn-graded algebra T is semi-commutative, then R is left and right finitely partitive. In order to do this, a multi-variable Poincaré series for every finitely generated graded T-module is considered and it is shown that this Poincaré series is a rational function. The methods apply to some iterated Ore extensions such as quantum matrices and quantum Weyl algebras as well as to the quantized enveloping algebra of [sfr ][lfr ](ν+1).