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
1 Answer
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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: ${n+1 \brack {n+1k}}$. There are plenty of articles that face the problem of giving estimations of these numbers out there.

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$\begingroup$ I'm pretty sure it is what I said above.. $\endgroup$ Nov 21, 2019 at 13:52

1$\begingroup$ 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 $nk$ distinct integers in $\{1,...,n\}$. Therefore the coefficient of $x^{nk}$ is the sum of all products of $k$ distinct integers which is ${n+1 \brack {nk+1}}$. $\endgroup$– DigerNov 21, 2019 at 13:58

$\begingroup$ Yes, you're right. I edit and fix the typo now. $\endgroup$ Nov 21, 2019 at 14:34