So the problem I'm working on is as follows:
Let $\lambda$ and $\mu$ be integer partitions, and let $\lambda^*$ and $\mu^*$ be their conjugates. By counting a set in two ways, prove $\sum_{i,j}\min\{\lambda_i,\mu_j\}=\sum_k\lambda_k^* \mu_k^*$, where $\lambda_k$ is the $k$th part of the partition $\lambda$ and $\lambda_k^*$ is the $k$th part of the conjugate $\lambda^*$.
I've got many drawings of Ferrers diagrams of some $\gamma^*$ where $\gamma^*_k=\lambda_k^*\mu_k^*$ which easily counts the right-hand side and I can see how it counts the minimum values on the left-hand side of the identity, but I can seem to justify why it works. Any thoughts?