Quasirandom permutations are characterized by 4-point densities

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.