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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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