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Let $v=(a_1,\dots,a_n)\in\mathbb{R}^n$ where the $a_i$ are distinct and positive. For $\sigma\in S_n$, let $\sigma(v)=(a_{\sigma(1)},\dots,a_{\sigma(n)})$. For any hyperplane $H$ through the origin, let $m_H$ be the number of $\sigma\in S_n$ such that $\sigma(v)\in H$. What is $\max_H m_H$?

I suspect that the answer is $(n-1)!$. Notice that if $\sigma\in S_{n-1}$, then the $(n-1)!$ vectors $\sigma(v)$ are all contained in the hyperplane $(\sum_{i=1}^{n-1}a_i)x_n=a_n\sum_{i=1}^{n-1}x_i$.

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  • $\begingroup$ Do you mean the maximum over all possible choices of $v$, or the maximum as a function of $v$? (Also, are your hyperplanes always through the origin?) $\endgroup$ – LSpice Nov 20 '18 at 0:33
  • $\begingroup$ Good point. I edited the question. $\endgroup$ – user131566 Nov 20 '18 at 1:01
  • $\begingroup$ If $n$ is even and $v=(1,2,\dots,n)$, then the condition $\frac{n-2}2(x_1+x_2)=(x_3+\dots+x_n)$, equivalent with $x_1+x_2=n+1$, is satisfied on $(n-2)!n$ points. $\endgroup$ – Ilya Bogdanov Nov 20 '18 at 10:35
  • $\begingroup$ Your edit still doesn't seem directly to address my question: are you looking for $\max_H m_H(v)$ as a function of $v$, or for $\max_v \max_H m_H(v)$? $\endgroup$ – LSpice Nov 20 '18 at 15:28
  • $\begingroup$ Understanding $\max_H m_H(v)$ would be ideal, but I would be happy even knowing the answer to $\max_v\max_H m_H(v)$. $\endgroup$ – user131566 Nov 20 '18 at 15:32

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