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Assume $A$ is a doubly stochastic matrix. I was only able to show if $A$ is a permutation matrix, then $\operatorname{per} (A) = 1$. I wonder how to prove conversely?

Rodrigo de Azevedo
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Luren
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    Note that there is a trivial bound $\textrm{per}(A)\leq\prod\limits_{k=1}^{n}\left(\sum\limits_{j=1}^{n}a_{kj}\right)$ (for matrices $A$ with nonnegative entries). Hence, for doubly stochastic matrix $A$ we have $\textrm{per}(A)\leq 1$. Then, you need to find out when does the equality occur. – richrow May 12 '20 at 15:33
  • could you provide more details? – Luren May 12 '20 at 20:05
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    Expand the right hand side and note that all terms of permanent are also appear in the right hand side. Hence, the equality occurs when all remaining terms are equal to 0. – richrow May 13 '20 at 11:31
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    @richrow You should post that as an answer. – coiso May 08 '25 at 14:08
  • @coiso I assume that now the full answer is in the linked (newer) question https://math.stackexchange.com/questions/5063254/if-a-stochastic-matrix-has-unit-permanent-is-it-a-permutation-matrix – richrow May 08 '25 at 14:26

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