论文标题
$ s_n $ -orbit的哪一小部分可以放在超平面上？
What fraction of an $S_n$-orbit can lie on a hyperplane?
论文作者
论文摘要
考虑$ \ Mathbb {r}^n $上的$ s_n $ - action，由定位坐标给出。本文解决了以下问题：计算$ \ max_ {v，h} | h \ cap s_nv | $ as $ h \ subset \ subset \ mathbb {r}^n $范围通过所有超越原点的范围，$ v \ in \ mathbb {r}^n $ ranges coled coledates in the Origin and $ v \ in \ mathbb in \ mathbb {r} x_i = 0 $。我们猜想，对于$ n \ geq3 $，答案是$（n-1）！$对于奇数$ n $，而$ n（n-2）！$甚至$ n $。我们证明，如果$ p $是$ p \ leq n $的最大素数，则$ \ max_ {v，h} | h \ cap s_nv | \ leq \ leq \ frac {n！} {p} $。特别是，这证明了当$ n $或$ n-1 $是主要的猜想。
Consider the $S_n$-action on $\mathbb{R}^n$ given by permuting coordinates. This paper addresses the following problem: compute $\max_{v,H} |H\cap S_nv|$ as $H\subset\mathbb{R}^n$ ranges over all hyperplanes through the origin and $v\in\mathbb{R}^n$ ranges over all vectors with distinct coordinates that are not contained in the hyperplane $\sum x_i=0$. We conjecture that for $n\geq3$, the answer is $(n-1)!$ for odd $n$, and $n(n-2)!$ for even $n$. We prove that if $p$ is the largest prime with $p\leq n$, then $\max_{v,H} |H\cap S_nv|\leq \frac{n!}{p}$. In particular, this proves the conjecture when $n$ or $n-1$ is prime.
