TITLE: What is the relationship between the Bell numbers, the Bell polynomials, and the partition numbers?
QUESTION [9 upvotes]: A friend of mine and I were wondering what relationship exists between the partition numbers $p_{n}$ and the Bell numbers $B_{n}$ (and also possibly the Bell polynomials $B_{n,k}(x_1,x_2,\dots,x_{n-k+1})$ )? Since integers are the decategorification of finite sets, it is appealing to think that there should be some relationship between the generating functions of the Bell numbers and the partition numbers. My friend pointed out the the equivalence classes of partitions of sets correspond to the conjugacy classes of the symmetric group $S_{n}$, but it was not clear how to convert that into a relationship between the  two types of partitions.
One definite historical flavored question arose in our minds: why is there such a disconnect between the study of integer partitions -- which leads to things like the Ramanujan-Rademacher formula, and involves lots of clever q-series manipulations, and the study of set partitions, which involves things like Stirling numbers, Bell numbers, Bell polynomials and whatnot. Her book about integer partitions apparently didn't mention the Stirling numbers at all.

REPLY [3 votes]: This doesn't answer the question, but I would add another example to the Bell number side: The Bell number of a graph is the number of set partitions of the vertices into independent sets, so the Bell number of the empty graph on $n$ vertices is  $B_n$. 
Winston Yang proved that the Bell number of a tree on $n+1$ vertices is $B_n$, and the Bell number of the  $1$-skeleton of a simplicial complex called a $k$-tree with $n+k$-vertices is also $B_n$.