Let $\mathbf{k}= \langle k_1,...,k_u \rangle$ denote cardinalities of $\sum_{i=1}^{u}k_i = n$ objects, where $k_i$ is the cardinality of the $i^{th}$ object. In how many ways can I partition this partition into two new partitions $\mathbf{k'}= \langle k'_1,...,k'_u \rangle$ and $\mathbf{k''}= \langle k''_1,...,k''_u \rangle$, such that $\sum_{i=1}^{u}k'_i = m$ and $\sum_{i=1}^{u}k''_i = n-m$. Can I have something more informative than the solutions of the following equations:
$$\forall i. k'_i + k''_i = k_i$$ $$\sum_{i=1}^{u}k'_i = m$$ $$\sum_{i=1}^{u}k''_i = n-m$$ $$\sum_{i}^{u}k_i = n$$