Dan, These are the Bell numbers, B(n). A000110 in the EIS, where you can find formulas and references to your heart's content. --Michael Kleber On 2/28/06, Schroeppel, Richard <rschroe@sandia.gov> wrote:
This message is from Dan Asimov, about 10 days ago. The mailing list software didn't like it for some reason. I did some heavy editing to unHTML it, so I may have mangled it. --Rich ----------------------------------------------
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. Ordinary partitions are of course equivalence classes of these "labeled partitions" (my term). ; E.g., for X_5:
ordinary partitions: # of corresponding labeled partitions -------------------- -------------------------- 1+1+1+1+1 = 1 1+1+1+2 C(5,2) = 10 1+2+2 5 * C(4,2)/2 = 15 1+1+3 C(5,3) = 10 1+4 C(5,1) = 5 2+3 C(5,2) = 10 5 C(5,5) = 1 ----- total: 52
Is there a formula for LP(n) ? What about an asymptotic formula?
--Dan
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