Bell Numbers
The Formula Used To Find
The nth Term Of The
Bell Numbers
Abstract
A pattern was discovered when elements in a set were rearranged as many ways as possible without repeating. This pattern is a sequence of numbers called Bell Numbers. In combinatorial mathematics, which is said to be the mathematics of the finite, the nth Bell number is the number of partitions of a set with n members. This find the number of different ways an element or elements can be rearranged. For example a 3 element set of {a,b,c,} can be partitioned in 5 different ways.
{ {a}, {b}, {c} }
{ {a , {b, c} }
{ {b}, {a, c} }
{ {c}, {a, b} }
{ {a, b, c} }
Problem Statement
The problem in this experiment was to find to a procedure or formula to find the nth bell number. To find the procedure/formula of the nth bell number, a list of bell numbers will be looked at and based off the relationship between the numbers a procedure/formula will be determined. A conjecture for this problem must include a factorial in the formula because bell numbers find all the partitions of a set meaning they have to be rearranged in each way much like how middle school questions ask “how many different ways can you rearrange 4 crayons”.
Solutions
One way to calculate Bell Numbers is by looking at Aitken’s array. The way this array is formulated is by starting with the number 1 as the first row. After, take the leftmost number from each row and place that as the first number in the row preceding it. To find the number that should be all the way to the left, add the number above and then placing that number to the right. Repeat these steps until there is 1 more number than there was on the the previous row. Each number on the right of each row is the bell number. The number of the row being nth term and the rightmost number being the value of the nth term.
Aitken’s Array
1
1 2
2 3 5
The nth Term Of The
Bell Numbers
Abstract
A pattern was discovered when elements in a set were rearranged as many ways as possible without repeating. This pattern is a sequence of numbers called Bell Numbers. In combinatorial mathematics, which is said to be the mathematics of the finite, the nth Bell number is the number of partitions of a set with n members. This find the number of different ways an element or elements can be rearranged. For example a 3 element set of {a,b,c,} can be partitioned in 5 different ways.
{ {a}, {b}, {c} }
{ {a , {b, c} }
{ {b}, {a, c} }
{ {c}, {a, b} }
{ {a, b, c} }
Problem Statement
The problem in this experiment was to find to a procedure or formula to find the nth bell number. To find the procedure/formula of the nth bell number, a list of bell numbers will be looked at and based off the relationship between the numbers a procedure/formula will be determined. A conjecture for this problem must include a factorial in the formula because bell numbers find all the partitions of a set meaning they have to be rearranged in each way much like how middle school questions ask “how many different ways can you rearrange 4 crayons”.
Solutions
One way to calculate Bell Numbers is by looking at Aitken’s array. The way this array is formulated is by starting with the number 1 as the first row. After, take the leftmost number from each row and place that as the first number in the row preceding it. To find the number that should be all the way to the left, add the number above and then placing that number to the right. Repeat these steps until there is 1 more number than there was on the the previous row. Each number on the right of each row is the bell number. The number of the row being nth term and the rightmost number being the value of the nth term.
Aitken’s Array
1
1 2
2 3 5