Solution:
Let $k, a_{1}, \ldots, a_{k}, b_{1}, \ldots, b_{k}$ be a solution. Then $b_{1}, b_{1}+b_{2}, \ldots, b_{1}+\cdots+b_{k}$ is just some increasing sequence of positive integers. Considering the $a_{i}$ as multiplicities, the $a_{i}$'s and $b_{i}$'s uniquely determine a partition of $7$. Likewise, we can determine $a_{i}$'s and $b_{i}$'s from any partition of $7$, so the number of solutions is $p(7)=15$.