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 2^{N-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.
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Re^3: Sets of subsets
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Definitely a good point. However, I'm sort of used to analyses sometimes taking upwards of days, so "blowing up" I guess might mean something else to me. You are correct though, very computationally complex. This was really just a random thought I had, I was wondering what sort of information I could extract out of this data. I imagine I'll end up going with a k-means clustering or something of that nature.