TITLE: Name of a particular "sum of products" function
QUESTION [1 upvotes]: Let $n$ and $\alpha$ be non-negative integers, let $[n] = \{1, \ldots, n\}$, and let $E$ be a subset of $[n]$. Define $S(n, \alpha)$ to be the sum of all products over subsets of size $\alpha$, or
$$ S(n,\alpha) = \sum_{|E| = \alpha} \prod_{m \in E} m.$$
What is the name of this function? Certainly it has a name and is well-documented, but searching fails me.

REPLY [1 votes]: That function outputs the "unsigned Stirling numbers of the first kind". Since that is already a very common name attached to a shifted/reflected variation of your function, there may not be another (commonly used) name. 
Define $c(n,k)$ to be the number of permutations of $\{1,\ldots,n\}$ with exactly $k$ cycles in its cycle decomposition. (This is a standard definition of the unsigned Stirling numbers of the first kind.) Then $S(n-1,n-k) = c(n,k)$. This is not immediately apparent from the definitions. There is a combinatorial proof given in Richard Stanley's Enumerative Combinatorics (Volume 1). It is the second proof of Proposition 1.3.4.
{"set_name": "stack_exchange", "score": 1, "question_id": 1155423}