For two distinguishable particles in a haromonic confinining potential in three dimensions, the imaginary time propagator is
(1)
The partition function is the trace of the propagator for ,
(2)
For identical particles, we need to symmetrize the states for fermions, or antisymmetrize the states for bosons. The trace of the permuted propagator is,
Next we change coordinates to and . Then and , and we find,
(3)