The total number of partitions of a k-element...

Q) The total number of partitions of a k-element
   set is denoted by B (k) and is called the k-th Bell number.

    Thus B (1) = 1and B (2) = 2.
   Need to find B (k) for k = 4, 5, 6.

Sol:

We know that,

Where,

Is the Binomial Coefficient.

It is given by:

It’s given that B (0) =1; B (1) =1; B (2) =2

Hence for n=3, from the first formulae

B (3) =C (2, K) B (K) where K=0 to n-1

B(3)=C(2,0)B(0)+ C(2,1)B(1)+ C(2,2)B(2)

From the second formulae, C (2, 0) =2! /0! (2-0) = 1

C (2, 1) =2; C (2, 2) =1

Hence,

B (3) =1*1+2*1+1*2=5

For n=4,

B(4)= C(3,0)B(0)+ C(3,1)B(1)+ C(3,2)B(2) + C(3,3)B(3)

        = 1*1+3*1+3*2+1*5= 15

Similarly,

For n=5,

B (5) = 52

For n=6,

B (6) = 203