# Adding a function of a subset over all possible subsets

So the context that I have for this problem comes from what I'm currently working on, but I can reduce the point that I'm stuck at to a more abstract problem. Since it stems from another problem, I can give more details about the nature of relevant terms if it simplifies anything.

Consider some set of positive numbers $X=\{x_1\cdots x_n\}$. Let $S_k$ be a $k$ element subset of $X$. Define $D_{S_k} = \sum_{x\in S_k} x$.

What I'm interested in is expressing $\sum_{S_k} f(D_{s_k})$ in terms of statistics of the set $X$ and the function $f$, where $\sum_{S_k}$ is to be interpreted as a sum over all possible $k$ element subsets of $X$.

Now, for simple functions like $f(x)=x$, I can show that $$\sum_{S_k} D_{s_k} = \sum_{S_k} k\langle X\rangle = \binom{n}{k}k\langle X \rangle$$ and similarly for $f(x)=x^2$ I can show that $\sum_{S_k} D_{s_k}^2 = \binom{n}{k}\times\left((k\langle X \rangle)^2 + k (\langle X^2 \rangle - \langle X \rangle^2)\right)$.

I can similarly solve for $f(x)=x^n$. Now where I'm stuck at, is in getting expressions for when $f(x)=1/x$ or $f(x)=1/x^2$. Of course if someone could tell me how to go about doing this for general $f$ that would be great, but right now just $f$ in these two forms could be pretty helpful.

If it helps, in my context $x_i\in \mathbb{N}$.

EDIT: So it turns out that just $f(x)=1/x$ and $f(x)=1/x^2$ wont suffice for my context, rather I'll need $f(x)=1/x^r$ for arbitrary $r\in \mathbb{N}$ for the purposes of my problem where I'm coming up with this. I guess that only makes things worse. Any ideas on how to proceed?

EDIT2: I have also posted this on Math Overflow here

EDIT3: Does anyone have any idea about possible directions or maybe other problems that are somewhat similar in nature to what I need?

• I don't have an answer, but it seems to me that this question has to do with symmetric polynomials. I've tagged it accordingly. – Michael Lugo Oct 24 '16 at 15:03
• @MichaelLugo: Sure, thanks. – SarthakC Oct 24 '16 at 19:07
• MO copy of the question: mathoverflow.net/questions/253894/… If you cross post a question, it is customary to add link to to the other copy into your post. See the advice given here, Other discussions about cross-posting might be of interest, too. – Martin Sleziak Nov 4 '16 at 6:26
• I was unaware of that. Thanks for informing me. I'll add a link to the other post in both of my posts. – SarthakC Nov 4 '16 at 14:10