I probably didn't phrase it very well. Really I'm just interested in the partitions of the main set of things. So if there were three "things", there would only be 4 sets of subsets:
Since ordering doesn't matter. From these partitions I can calculate the distance or similarity between the groups from the attribute table (which has already been calculated). So for the above example I might have a table like:
t1 t2 t3
t1 0 1 2
t2 1 0 3
t3 2 3 0
Representing the difference between each thing. When I have the subsets from above I can calculate what the average difference is between the subsets using the difference values as the metric.
Important to note as well that the number of "things" themselves are relatively few. The attributes previously calculated are in the hundreds, but the things themselves number a few tens (max 24). So, I'm not too worried about the numbers exploding on me... yet.