Quasirandom permutations are characterized by 4-point densities

TL;DR

This paper proves that permutations with 4-point densities approaching uniformity are quasirandom, providing a characterization based on 4-point subpermutation densities, thus answering a longstanding question.

Contribution

It establishes that 4-point densities uniquely characterize quasirandom permutations, resolving a question posed by Graham.

Findings

Sequences with 4-point densities approaching 1/24 are quasirandom.

Characterization of quasirandom permutations via 4-point densities.

Answers a previously open question in permutation theory.

Abstract

For permutations P and T of lengths |P|\le|T|, let t(P,T) be the probability that the restriction of T to a random |P|-point set is (order) isomorphic to P. We show that every sequence \{T_j\} of permutations such that |T_j|\to\infty and t(P,T_j)\to 1/4! for every 4-point permutation P is quasirandom (that is, t(P,T_j)\to 1/|P|! for every P). This answers a question posed by Graham.