A formula on partitions Suppose that $\lambda,\mu$ are integer partitions, with conjugates $\lambda^*,\mu^*$. Could you help me to prove the following formula, please?

$\sum_{i,j}\mathrm{min}(\lambda_i,\mu_j)=\sum_k\lambda^*_k\mu^*_k$


The identity describes two ways of counting the cubic ‘cells’ in a plane partition:
Let $n_{i,j} = \min(\lambda_i,\mu_j)$ be a plane partition, then the $k$th ‘storey’ is rectangular with dimensions $\lambda^\star_k\times\mu^\star_k$:
The cells in the $k$th storey are the $(i,j)$ for which both $\lambda_i\geqslant k$ and $\mu_j\geqslant k$. But $\lambda^\star_k$ is the number of $i$ for which $\lambda_i\geqslant k$, and $\mu^\star_k$ is the number of $j$ for which $\mu_j\geqslant k$.
