# How Uniform is the Uniform Distribution on Permutations?

## TL;DR

This paper investigates whether the uniform distribution on permutations approximates the uniform distribution on a sphere, finding that permutations are highly localized and do not evenly cover the sphere as size increases.

## Contribution

It demonstrates that permutations do not uniformly cover the sphere, contrasting with initial intuition, and explores the geometric properties of permutation sets on spheres.

## Key findings

- Permutations are confined to a negligible portion of the sphere.
- The permutohedron occupies a negligible volume of the ball.
- Largest empty spherical cap size approaches zero for favorable configurations.

## Abstract

For large $q$, does the (discrete) uniform distribution on the set of $q!$ permutations of the vector $(1,2,\dots,q)$ closely approximate the (continuous) uniform distribution on the $(q-2)$-sphere that contains them? These permutations comprise the vertices of the regular permutohedron, a $(q-1)$-dimensional convex polyhedron. Surprisingly to me, the answer is emphatically no: these permutations are confined to a negligible portion of the sphere, and the regular permutohedron occupies a negligible portion of the ball. However, $(1,2,\dots,q)$ is not the most favorable configuration for approximate spherical uniformity of permutations. Unlike the permutations of $(1,2,\dots,q)$, the normalized surface area of the largest empty spherical cap among the permutations of the most favorable configuration approaches 0 as $q\to\infty$. Several open questions are posed.