# Sum of all products of k distinct integers in [1,n] [duplicate]

Let $$S=\{1,2,3,...,n\}$$ be the set of integers up to $$n$$ and $$p_k(a_1,...,a_k)=a_1\cdots a_k$$ the product of $$k$$ distinct integers $$a_1,...,a_k \in S$$. There are $$\binom{n}{k}$$ possibilities to construct such a product $$p_k$$. I was wondering if it is anyhow possible to estimate the sum of all such $$k$$-products $$p_k$$ or similarly the mean value as $$n$$ is large. Any idea? Thanks

• Try induction over $n$. Nov 21, 2019 at 8:53

The sum of all $$k$$-products of numbers in the interval $$\{1,2,\ldots,n\}$$, it is, the number you are referring to, is known as the Stirling number of the first kind: $${n+1 \brack {n+1-k}}$$. There are plenty of articles that face the problem of giving estimations of these numbers out there.
• Do you mean ${n+1 \brack {n-k+1}}$ ? Nov 21, 2019 at 11:48
• Hm, but $(x+1)\cdots(x+n)=\sum_{k=0}^n {n+1 \brack {k+1}} \, x^k$ and the coefficient of $x^k$ is the sum of all products of $n-k$ distinct integers in $\{1,...,n\}$. Therefore the coefficient of $x^{n-k}$ is the sum of all products of $k$ distinct integers which is ${n+1 \brack {n-k+1}}$. Nov 21, 2019 at 13:58