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

$sum_i,jmathrmmin(lambda_i,mu_j)=sum_klambda^*_kmu^*_k$

This is really the same answer as David Bevan's, but formulated a bit differently. Both sides of the equation count triples $(i,j,k)$ where $kleqlambda_i$ and $kleqmu_j$. If you fix $i$ and $j$ you have $min(lambda_i,mu_j)$ different choices for $k$, while if you fix $k$ you have $lambda^*_k$ choices for $i$ and independently $mu^*_k$ choices for $j$.

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 $lambda^star_ktimesmu^star_k$:

The cells in the $k$th storey are the $(i,j)$ for which both $lambda_igeqslant k$ and $mu_jgeqslant k$. But $lambda^star_k$ is the number of $i$ for which $lambda_igeqslant k$, and $mu^star_k$ is the number of $j$ for which $mu_jgeqslant k$.

I think induction on the integer that $mu$ partitions works. Subtract 1 from the smallest part of $mu$, and show that both sides decrease by the same amount?

Suppose two finite integer non negative sequences $t$ and $s$; let min($t,$s) be the sequence $[min(t_i,s_j)]$, reordered anyway.
It is immediate that $min(t_i,s_j)=> k$ is equivalent to $t=> k$ and $s=> k$; hence the conjugate dual sequence is the following: $s*.t*$.
The result follows since we know that the sum of a finite, integer non negative sequence and the sum of its conjugate are equal.