Hah! I hadn't even noticed my post hadn't gone through. After posting I found a recursion formula & calculated the terms up through 8 by hand, which were enough numbers to find the sequence in OEIS. (So I wrote a follow-up post saying Never mind, I learned that they're Bell numbers. Apparently that post is stuck, too.) And Jim, thanks for recommending the Rota book, but if you happen to know an asymptotic formula I'd love if you'd post it. --Dan I wrote about 10 days ago: << Suppose you want to count all possible partitions of the set X_n = {1,2,...,n}. I.e., you want to know how many [collections of mutually disjoint nonempty subsets of X_n] have union = X_n. . . .
Michael Kleber wrote: << These are the Bell numbers, B(n). A000110 in the EIS, where you can find formulas and references to your heart's content.