The fraction of an $S_n$-orbit on a hyperplane

TL;DR

This paper proves a conjecture about the maximum number of permuted vectors of a distinct-coordinate vector that can lie on a hyperplane, refining understanding of symmetry and permutation group actions.

Contribution

It provides a proof for Huang, McKinnon, and Satriano's conjecture on the maximum fraction of an $S_n$-orbit on certain hyperplanes.

Findings

The conjecture is proven for all $n \

Maximum orbit fraction is bounded by $2\lfloor n/2 \rfloor (n-2)!$

Clarifies the structure of permutation orbits on hyperplanes.

Abstract

Huang, McKinnon, and Satriano conjectured that if $v \in \mathbb{R}^n$ has distinct coordinates and $n \geq 3$, then a hyperplane through the origin other than $\sum_i x_i = 0$ contains at most $2\lfloor n/2 \rfloor (n-2)!$ of the vectors obtained by permuting the coordinates of $v$. We prove this conjecture.