TITLE: Sum over integer compositions
QUESTION [5 upvotes]: 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

REPLY [3 votes]: Yes, there are closed-form expressions. The number $S(n,k)$ you are looking for is the number of weighted integer compositions with weighting function $f(a)=a^p$. Many recursions and other representations for this number exist.
For example, the number $S(n,k)$ from above is precisely the number $d_{S,f}(n,k)$ defined in Eger (2013), Restricted weighted integer compositions and extended binomial coefficients (for $S=\{1,2,3,…,\}$ and $f(a)=a^p$). This paper says that $S(n,k)$ is an extended binomial coefficient, and gives various representations of the extended binomial coefficients. Other relevant literature would be Fahssi (2012), The polynomial triangles revisited, and Shapcott (2013), C-color compositions and palindromes. More relevant literature can be found in the references of these works. Other work of C. Shapcott also addresses part-products of integer compositions, which is related to your case $p=1$.