  We study the moduli space of $SU(4)$ invariant BPS conditions in
supersymmetric gauge theory on non-commutative ${\mathbb C}^4$ by means of an
ADHM-like quiver construction and we classify the invariant solutions under the
natural toric action in terms of solid partitions. In the orbifold case
${\mathbb C}^4/G$, $G$ being a finite subgroup of $SU(4)$, the classification
is given in terms of coloured solid partitions. The statistical weight for
their counting is defined through the associated equivariant cohomological
gauge theory. We explicitly compute its partition function on ${\mathbb C}^4$
and ${\mathbb C}^2\times\left({\mathbb C}^2/{\mathbb Z}_2\right)$ which
conjecturally provides the corresponding orbifold Donaldson-Thomas invariants.
