  We study generating functions of ordinary and plane partitions coloured by
the action of a finite subgroup of the corresponding special linear group.
After reviewing known results for the case of ordinary partitions, we formulate
a conjecture concerning a factorisation property of the generating function of
coloured plane partitions that can be thought of as an orbifold analogue of a
conjecture of Maulik et al., now a theorem, in three-dimensional
Donaldson-Thomas theory. We study natural quantisations of the generating
functions arising from geometry, discuss a quantised version of our conjecture,
and prove a positivity result for the quantised coloured plane partition
function under a geometric assumption.
