Perfect Partitions

In your house, you have a lot of integers. These integers are coloured by $Q$ colours (numbered $1$ through $Q$). For each valid $i$, you have $b_i$ integers with colour $i$ and each of these integers has value $a_i$. For each integer $i$ such that $1 \le i \le N$, you need to find the number of different partitions of $i$. Here, a partition of $i$ consists of one or more integers from your hou