Number of bells (from Wikipedia) & Stirling

Bell number

The Bell number , named Eric Tampur Bell (Eric Temple Bell), is a series of integers in combinatorial mathematics, with the opening being (the A000110 sequence of Oeis):

Bell number

B n is the number of partitioning methods for the set of cardinality n . A division of Set s is a family of 22 disjoint non-empty sets defined as s , and they are s. For example, B3 = 5 because the collection of 3 elements {A, B, c} has 5 different partitioning methods:

{{ a}, { b}, { c}}

{{ a}}, { b , c}}

{{ b}, { a, c}}

{{ c}, { a, b}}

{{ a, b, c}};

B0 is 1 because the empty set has exactly 1 methods of partitioning. Each member of the empty set is a non-empty collection (This is vacuous truth, because the empty sets do not actually have members), and they are empty. So the empty set is its only division.

The bell number fits the recursive formula:

Proof of the above combinatorial formula:

Think of it this way, b_{n+1} is the number of partitions that contain a collection of n+1 elements, considering the elements

Assuming that he is divided into one category alone, then there are still n elements, in which case the number is divided;

Assuming that he and an element are divided into a class, then there are n-1 elements left, in which case the number is;

Assuming that he and some two elements are divided into a class, then there are still n-2 elements, in this case the number is divided;

And so on, we get the combination formula

They are also suitable for the "Dobinski formula":

the n -th moment of the Poisson fraction with expected value of 1.

They are also suitable for "Touchard": if p is any prime number, then

Each bell number is the "second class Stirling number" and

The number of Stirling S(n, K) is the number of methods that divide the set of cardinality n into exactly K non-empty sets.

The polynomial of the n -th moment of any probability distribution in the first n cumulant, the coefficients and the nth bell numbers. This method of dividing numbers is not as coarse as the method of Stirling.

The exponential parent function of the bell number is

Bell Triangles [edit ]

Construct a triangular matrix (in the form of a Yang Hui triangle) using the following method:

• The first line of the first row is 1 ()
• For N>1, the first of the nth rows is equivalent to the last item of line n-1. （）
• For m,n>1, the nth row of item m equals the sum of two numbers on the left and upper left. （）

The results are as follows: (oeis:a011971)

Each line entry is a bell number. The sum of each row is the second class of Stirling.

This triangle is called the Bell Triangle, the Aitken array, or the Peirce triangle (Bell Triangle, Aitken ' s array, Peirce triangle).

See also [edit ]

• Bell polynomial

references [edit ]

• http://planetmath.org/?op=getobj&from=objects&id=9059

Classification:

• Integer series

Number of bells (from Wikipedia) & Stirling