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

Read more about this topic:  Bell Number

Famous quotes containing the words scheme, calculating, bell and/or numbers:

    “We doubt not the destiny of our country—that she is to accomplish great things for human nature, and be the mother of a nobler race than the world has yet known. But she has been so false to the scheme made out at her nativity, that it is now hard to say which way that destiny points.”
    —Margaret Fuller (1810–1850)

    “Because relationships are a primary source of self-esteem for girls and women, daughters need to know they will not lose our love if they speak up for what they want to tell us how they feel about things. . . . Teaching girls to make specific requests, rather than being indirect and agreeable, will help them avoid the pitfalls of having to be manipulative and calculating to get what they want.”
    —Jeanne Elium (20th century)

    “In 1862 the congregation of the church forwarded the church bell to General Beauregard to be melted into cannon, “hoping that its gentle tones, that have so often called us to the House of God, may be transmuted into war’s resounding rhyme to repel the ruthless invader from the beautiful land God, in his goodness, has given us.””
    —Federal Writers’ Project Of The Wor, U.S. public relief program (1935-1943)

    “Think of the earth as a living organism that is being attacked by billions of bacteria whose numbers double every forty years. Either the host dies, or the virus dies, or both die.”
    —Gore Vidal (b. 1925)