Bell Numbers

     Bell Numbers, or Bell's Numbers, are the sequence {1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147....}.  The numbers count the ways that N distinguishable objects can be grouped into sets if no set can be empty.  For example the letters ABC can be grouped into sets so that:

1)   A, B, and C are in three seperate sets;
2)  A and B are together and C is seperate;
3)  A and C are together and B is seperate;
4)  B and C are together and A is seperate;
5)  or A, B, and C are all together in a single set.
 Thus when N=3, there are five partitions, so the third Bell number is 5.

     The Bell numbers are also the coefficients of the Maclauren expansion of

            =    

 

The numbers can be constructed by using the Bell Triangle, a name suggested to Martin Gardner by Jeffrey Shallit.  Start with a row with the number one.  Afterward each row begins with the last number of the previous row and continues to the right adding each number to the number above it to get the next number in the row..
 

1                           Start with one
1    2                     Start with one add 1+1 to get 2
2    3   5                Start with two, add 2+1=3, 3+2=5
5    7  10  15          Start with five, add 5+2=7, 7+3=10, 10+5=15...  etc...
15 20 27  37   52
52...........

Notice that the Bell numbers appear down the left column of the page...
You can find more about the Bell numbers at this Math Forum web page.
The Bell numbers are named for mathematician and writer  Eric Temple Bell (1883-1960) because he was one of the first to do in-depth analysis of the sequence.  The mathematician John Riordan was the first to use B(n) in honor of Bell.  Bell was a contributor to number theory, but is better known for his many books on the History of Mathematics.  Men of Mathematics was the first math history book I ever encountered, and his stories of the lives of mathematicians colored my study and teaching of mathematics.  Bell also was a prolific writer of science fiction under the name John Taine.