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 partitions (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 partition, but this seems quite weak as a bound.

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

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 partitions (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 $p=-2$.

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

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 partitions (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 partition, but this seems quite weak as a bound.

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 partitions 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 $p=-2$.

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

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 partitions (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 $p=-2$.

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