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## New answers tagged permutations

4

For an $n \times m$ matrix $M$, define the potential function $$\Phi(M) = \sum_{i=1}^n \sum_{j=1}^m 2^{n-1-i} 2^{m-1-j} M(i,j).$$ If we write it as $$\Phi(M) = \sum_{i=1}^n 2^{n-1-i} \sum_{j=1}^m 2^{m-1-j} M(i,j)$$ then it immediately follows that reordering rows either fixes $M$, or strictly increases $\Phi$. Similarly, reordering columns either fixes \$...

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