Let $n$ be a positive integer. For each $k = 1, \ldots, n$, let
$$
S\_k(n) := \sum\_{1 \le i\_1 < \cdots < i\_k \le n} i\_1 \cdots i\_k
$$
be the sum of the $k$-wise product of distinct integers from $1$ to $n$. Does this sum $S\_k(n)$ have a name and formula?


For example, $$S\_1(n) = 1 + \cdots + n = \binom{n + 1}{2}$$ and $$S\_2(n) = [(1 + \cdots + n)^2 - (1^2 + \cdots + n^2)]/2 = n(n+1)(3n^2 - n - 2)/24.$$



Name of the sum of the $k$-wise product of distinct integers from $1$ to $n$
These are the [Stirling numbers of the first kind](https://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind). They enumerate, among other things, the number of permutations with a given number of cycles.


