Forcing quasirandomness with 4-point permutations

TL;DR

This paper investigates the minimal size of permutation sets that enforce quasirandomness, proving that any such set must contain at least five 4-point permutations, advancing understanding of permutation quasirandomness.

Contribution

It establishes a lower bound of five on the size of quasirandom-forcing sets of 4-point permutations, improving previous knowledge of their minimal size.

Findings

Sets of fewer than five 4-point permutations cannot be quasirandom-forcing.

The known quasirandom-forcing sets of size eight are not minimal.

The result breaks the linear dependency barrier in perturbation gradients.

Abstract

A combinatorial object is said to be quasirandom if it exhibits certain properties that are typically seen in a truly random object of the same kind. It is known that a permutation is quasirandom if and only if the pattern density of each of the twenty-four 4-point permutations is close to 1/24, which is its expected value in a random permutation. In other words, the set of all twenty-four 4-point permutations is quasirandom-forcing. Moreover, it is known that there exist sets of eight 4-point permutations that are also quasirandom-forcing. Breaking the barrier of linear dependency of perturbation gradients, we show that every quasirandom-forcing set of 4-point permutations must have cardinality at least five.