Sorry if the question is trivial - are there closed form expressions or good approximations for the sum of a symmetric function taken over all integer compositions (into given number of parts) of a number?


More precisely, I'm interested in: 
$$
S(n,k) = \sum\_{a1+ \cdots +a\_k = n, \ \ a\_i \geq 1} \phi\_k(a\_1,\dots,a\_k)
$$


where $\phi\_k = \prod\_i a\_i^p$, e.g. for $p=-2$, but I'm curious even about $p=1$


I realize that I can bound this (using AM-GM ineq.) by replacing all terms by the most (un)balanced composition, but this seems quite weak as a bound.


EDITED: partition composition



Sum over integer compositions
Since the question is about compositions, there is a generating function approach when $\phi(a\_1,\ldots,a\_n)=\prod\_i f(a\_1)$ for some function $f$. Namely, $S(n,k)$ is the coefficient of $x^n$ in $\left(\sum\_{i\ge 1} f(i)x^i\right)^k$. For example if $f(i)=i$ then $\sum\_{i\ge 1} f(i)x^i=x/(x-1)^2$ so $S(k,n)$ is the coefficient of $x^n$ in $x^k/(1-x)^{2k}$, which is $$\binom{n+k-1}{2k-1}$$ if I got it right. Other small powers can be done the same way. More complicated $f$ are likely to be able to be handled asymptotically by standard analytic methods.


