Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

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

share|improve this question
add comment

3 Answers 3

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 $k\leq\lambda_i$ and $k\leq\mu_j$. 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 $j$.

share|improve this answer
add comment

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 $\mu_j\geqslant k$.

share|improve this answer
    
how can i prove that the $k$th storey is rectangular with dimensions $\lambda_k^*\mu^*_k$? I'm finding some difficulties –  Alex M Nov 11 '11 at 19:47
    
Brief explanation added above. –  David Bevan Nov 12 '11 at 9:57
add comment

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?

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.