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For a n-element set, how many ways to divide the set to a combination of subsets?

For example, for the set {1, 2, 3}, there would be 5 ways to divide:

{1, 2, 3}
{1}, {2, 3}
{2}, {1, 3}
{3}, {1, 2}
{1}, {2}, {3}

For the n-element set, I thought it would be

(n^n)/n!

, because imagine dividing as drop n balls to n indistinguishable boxes. But apparently, it is incorrect in the case n=3 .

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1 Answer

up vote 8 down vote accepted

You want the Bell numbers --- link to Wikipedia entry.

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Thanks a lot. Sorry, I'm a new user in the math stackexchange, so I can't vote up your answer. –  Sapience Oct 9 '12 at 1:15
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