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Bell numbers

Bell numbers: counting the partitions of a give set
Now, we just computed the number of n partitions of Ak, but how many are all of the partitions of Ak. For instance, we want to subdivide 14 students into 3 study groups, but with possibly empty study groups. So they may be subdivided into 2 groups and that’s all or one group and that’s all. Well, of course, this is the sum of the stirling number of second kind, 14/1, plus stirling number of second kind, 14/2, plus 14/3. And this is called the third Bell number B3.
More precisely, if k is an integer greater or equal than 1, the Bell number, number k, Bk, is the number of all possible partitions of Ak. So alternatively, it is the sum of the stirling number of second kind, k/1, k/2, k/k.
Let us look, for instance, at all the partitions of I5.
Well, this is taken from the user Watchduck of Wikimedia Commons. There is the unique 1 partition of I5. There are the 2 partitions of I5, the 3 partitions of I5, the 4 partitions of I5, and the unique 5 partitions of I5. And this leaves all the 52 partitions of I5.

The number of partitions that can be formed with a set containing a given number (k) of elements is the (k)-th Bell number. Let us discover how it is related with the Stirling numbers of the second kind and how to compute it.

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Combinatorics: Strategies and Methods for Counting

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