Partition of a number with some condition

Question

Given two positive numbers $r,s$, what is a positive solution of $$\sum_{i=0}^{m-1} k_i = r \quad\text{ and } \quad \sum_{i=0}^{m-1} (i+1)k_i = s \quad ??$$

I would like to get some reference to treat this kind of partition.

remark: There is a famous related subject called "Ewens's sampling formula".

Answer 1

This is a partition of $s$ into $r$ non-zero parts, with $k_i$ parts $i+1$. It would seem slightly more straightforward to write it as

$$ \sum_{i=1}^mk_i=r\quad\text{and}\quad\sum_{i=1}^mik_i=s $$

with $k_i$ parts $i$.