Statistics How To

Bell’s Numbers and the Bell Triangle

Statistics Definitions > Bell’s numbers

Bell’s Numbers and the Bell Triangle (sometimes called the Pierce triangle or Aitken’s array) are a sequence of numbers which count the possible partitions of a set, and the triangle which makes derivation of them easy.

Bell’s Numbers: What they Are and What they Mean

The first Bell numbers are:

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975.

The nth Bell number, Bn, is the number of nonempty subsets a set of size n can be partitioned into. B0 is defined as one (i.e. the first number in the above list); There is just one possible partition of the set containing one member, so B1 = 1.

There are two partitions possible for a set with two elements, so B2 is just two.

A set with three elements can be partitioned five ways. Taking the set {a, b, c}, the five possible partitions are:

  • {(a) (b) (c)},
  • {(a, b), (c)},
  • {(a,c) (b)},
  • {(b,c), a},
  • {(a, b, c)}.

There isn’t any simple formula that can give us Bn, but we can find Bell’s numbers in the Bell Triangle, in the next section, or use the following recursive equation to define them:

bell's numbers

Bell’s Numbers and the Bell Triangle as a Way to Derive them

The Bell triangle is an easy-to-fill in right triangle which gives us, in the left hand column, all of the Bell numbers.

1. Creating the Triangle
We make the triangle this way:

  • On row one, write the number 1
  • Begin all other rows with the last number of the previous row. The last number in row 1 was 1, so row 2 also begins with 1.
  • All other numbers are found by adding the last number to the one above it. To find out what to write in the second place in row 2, we look at the place before it, place 1. This is just 1, and the digit 1 is above it, so 1 + 1 = 2

The first part of the Bell triangle will therefore be

1
1 2

Following those same rules, we begin the third row with the last number in the previous row, or 2. Then we add that 2 to the number above it to find the next number is 3. Three plus the two above it makes five, so the row finishes with a five.

1
1 2
2 3 5

Row four starts with the last of row three, i.e., 5. Then 5 + 2 = 7, so the next place is 7, and 7 + 3 = 10, so the next digit is ten. Finishing this row and the next in the same manner, we get

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

For any row k, the number that starts the row is Bk-1. So the first digit of row 1 is B1 -1 = B(0), the first digit of row 4 is B(3), and the first digit of row 55 will be B(54).

2. Reading the Triangle
The Bell numbers are given by the far left column, bolded here:
1
1 2
2 3 5
5 7 10 15
15 20 27 37 52

Note that the triangle is sometimes mirrored, so Bell’s numbers appear in the right column, and not the left; Make sure you know the construction of the triangle before attempting to read it.

References

Guichard, D. Combinatorics and Graph Theory. Retrieved from
https://www.whitman.edu/mathematics/cgt_online/book/section01.04.html on February 4, 2018
Weisstein, Eric W. “Bell Triangle.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/BellTriangle.html

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