How Uniform is the Uniform Distribution on Permutations?

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.

37 pages

References in corpus (1)

• Sharp bounds for harmonic numbers