Let $(a_1^{r_1},\ldots,a_{p}^{r_{p}})\vdash n$ be the multiplicity representation of an integer partition of n. Each $a_{i}$ is a part of the partition and $r_{i}$ is its corresponding size. We exclude sizes of 0. Thus, for example, $(1^{4},3^{2},6^{1})$ is the multiplicity representation of the partition $(1,1,1,1,3,3,6)$ of 16. Consider the following sum which sums over every partition of n (so p is not constant – f is merely a function over the multiset of sizes for each partition)...
$$ \sum_{(a_1^{r_1},\ldots,a_{p}^{r_{p}}) \vdash n} f(r_1,...,r_p)$$
Since the summand only refers to the sizes $r_{i}$ of the parts and does not refer to the parts $a_{i}$ themselves, this seems like an inefficient way of expressing and computing the sum.
Can you recommend a better, more efficient way of expressing such a sum? In particular, can you recommend a sum where what is being summed over does not refer to the parts of the partitions?
Note: Summing over the solutions to the corresponding Diophantine equation still refers to the parts.