Characterization of quasirandom permutations by a pattern sum

TL;DR

This paper characterizes quasirandom permutation sequences through specific pattern sum conditions, identifying exactly ten sets of 4-point permutations that serve as such characterizations.

Contribution

It introduces a novel pattern sum criterion for quasirandomness and fully characterizes the sets of permutations that satisfy this property.

Findings

Exactly ten sets of 4-point permutations characterize quasirandomness.

The smallest such set has eight permutations.

Pattern sum convergence is equivalent to quasirandomness for these sets.

Abstract

It is known that a sequence Pi_i of permutations is quasirandom if and only if the pattern density of every 4-point permutation in Pi_i converges to 1/24. We show that there is a set S of 4-point permutations such that the sum of the pattern densities of the permutations from S in the permutations Pi_i converges to |S|/24 if and only if the sequence is quasirandom. Moreover, we are able to completely characterize the sets S with this property. In particular, there are exactly ten such sets, the smallest of which has cardinality eight.