Bell Number - Triangle Scheme For Calculating Bell Numbers

Triangle Scheme For Calculating Bell Numbers

The Bell numbers can easily be calculated by creating the so-called Bell triangle, also called Aitken's array or the Peirce triangle:

1. Start with the number one. Put this on a row by itself.
2. Start a new row with the rightmost element from the previous row as the leftmost number
3. Determine the numbers not on the left column by taking the sum of the number to the left and the number above the number to the left (the number diagonally up and left of the number we are calculating)
4. Repeat step three until there is a new row with one more number than the previous row
5. The number on the left hand side of a given row is the Bell number for that row.

For example, the first row is made by placing one by itself. The next (second) row is made by taking the rightmost number from the previous row (1), and placing it on a new row. We now have a structure like this:

1 1 ''x''

The value x here is determined by adding the number to the left of x (one) and the number above the number to the left of x (also one).

1 1 2 y

The value y is determined by copying over the number from the right of the previous row. Since the number on the right hand side of the previous row has a value of 2, y is given a value of two.

1 1 2 2 3 ''x''

Again, since x is not the leftmost element of a given row, its value is determined by taking the sum of the number to x's left (three) and the number above the number to x's left (two). The sum is five.

Here is the first five rows of this triangle:

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

The fifth row is calculated thus:

• Take 15 from the previous row
• 15 + 5 = 20
• 20 + 7 = 27
• 27 + 10 = 37
• 37 + 15 = 52