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 $(\sum_{i=1}^{n-1}a_i)x_n=a_n\sum_{i=1}^{n-1}x_i$.