Just look at the group t1 is in. t1 can be a group in itself. Or with each of t2, t3, ..., tk. Or any combination of them. Which means that if you have N "things", there are 2N-1-1 ways to be t1 in a group with other "t's" (-1 comes from that you seem to exclude the subset of all things together). And that's not even counting the different ways you can split up the group of things that aren't in the subset t1 belongs to.
in reply to Re^2: Sets of subsets
in thread Sets of subsets