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Bell number

The number of ways that n distinguishable objects (such as differently colored balls) can be grouped into sets (such as buckets) if no set can be empty. For example, if there are three balls, colored red (R), green (G), and blue (B), they can be grouped in five different ways: (RGB), (RG)(B), (RB)(G), (BG)(R), and (R)(G)(B), so that the third Bell number is 5. The sequence of Bell numbers, 1, 2, 5, 15, 52, 203, 877, 4140, 21147,..., can be built up in the form of a triangle, as follows. The first row has just the number one. Each successive row begins with the last number of the previous row and continues by adding the number just written down to the number immediately above and to the right of it.

1   2
2   3   5
   5    7  10  15
 15  20  27  37  52
52 ...                            

The Bell numbers appear down the left-hand side of the triangle. These normal Bell numbers contrast with ordered Bell numbers, which count the number of ways of placing n distinguishable object (balls) into one or more distinguishable sets (buckets) The ordered Bell numbers are 1, 3, 13, 75, 541, 4683, 47293, 545835, ...

Bell numbers, named after Eric Temple Bell, who was one of the first to analyze them in depth, are related to the Catalan numbers.

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