Quasirandom permutations are characterized by 4-point densities

Abstract

For permutations π and τ of lengths {pipe}π{pipe}≤{pipe}τ{pipe}, let t(π,τ) be the probability that the restriction of τ to a random {pipe}π{pipe}-point set is (order) isomorphic to π. We show that every sequence {τj} of permutations such that {pipe}τj{pipe} → ∞ and t(π,τj)→ 1/4! for every 4-point permutation π is quasirandom (that is, t(π,τj)→ 1/{pipe}π{pipe}! for every π). This answers a question posed by Graham. © 2013 The Author(s).