Question

Let Bn be the number of equivalence relations on the set n. Prove that Bn = Bn-k k-1

Answer 1

Solution:
Let $B_n$ be the number of partitions or equivalence relations possible in a set of elements. We now present a recurrence relation which will count the number $B_n$. The number $B_n$ is popularly known as Bell's number. Consider a partition of elements, say for example, .

In general
AP} P = {Ai, A2 ,
and
4, C 1,.,n)
and each $A_i$ are distinct. In
AP} P = {Ai, A2 ,
, consider $A_i$ containing the element and assume that $k=|A_i-\{n\}|$ . These elements of $A_i$ can be any subset in
1, n -1)
. Therefore, the number of such possible sets for $A_i$ is $\binom{n-1}{k}$, where $k\in\{1, . . . , n-1\}$ . Now to establish a recursive relation we focus on the remaining elements which are distributed among  
Ai . Ai-1 . A2+1 Ari
Interestingly, there are $B_{n-k-1}$ partitions among the remaining elements. Consequently, we get  
Bn-k-1 Bn-k k-1
.