<s>[INST] ### Bell Numbers as the Numbers of Partitions of a Set

The equalities
$B_{n} = \sum_{k=0}^{n} S(n,k)$ 
and
$S(n,k) = \frac{1}{k!} \sum_{i=0}^{k} \binom{k}{i} (-1)^{i} (k-i)^{n}$
are seen as the unification of the two concepts of Bell numbers that we previously mentioned, 
those involving set partitions, and those involving a sum of Stirling numbers of the second kind.

Here, we take a probabilistic approach of thinking about Bell numbers and its different aspects.
We define $P(n)$ and $S(n)$, as the permutation operator for $n$ objects and the sum operator, respectively.

Suppose the random variables $X$ and $Y$ have the permutation operator $P(n)$ and sum operator $S(n)$, respectively. If $E(X) = B_{n}$ and $E(Y) = B_{n}$, how do we go about determining 
$n$ such that  P\{X=k\} = P\{Y=k\} \ Walton2017?[/INST]