The Maximum of $\operatorname{per}(I-A)$ in Odd Order

Abstract: Let $Ω<em>n$ denote the set of $n\times n$ doubly stochastic matrices. Kim and Roush conjectured in 1981 that, for $n=2k+1&gt;1$, $$\max</em>{A\inΩ<em>{2k+1}}\operatorname{per}(I-A)=3\cdot 2<sup>{k-2}$$. They proposed the block construction $$A</em>\star=\frac12(J_3-I_3)\oplus P_2<sup>{\oplus(k-1)}$$, where $P_2=\begin{pmatrix}0&amp;1\1&amp;0\end{pmatrix}$. Here $J_3$ is the $3\times3$ all-ones matrix. They did not claim uniqueness. We fully prove their conjecture and classify equality: the maximizers are exactly the simultaneous-permutation conjugates of $A_\star$.