Combinatorial representation of function

I'll rename all three of your variables; you are asking for the number of partitions of $k$ that fit into an $m \times n$ box. This is famously known to be the coefficient of $q^k$ in the $q$-binomial coefficient

$${m+n \choose m}_q = \frac{[m+n]_q!}{[m]_q! [n]_q!}$$

where $[n]_q! = \prod_{i=1}^n \left( \frac{q^i - 1}{q - 1} \right)$ is the $q$-factorial. Much is known about these, including a $q$-analog of Pascal's identity that leads to a recurrence.