Characterization of quasirandom permutations by a pattern sum
Combinatorics
2022-07-18 v7
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.
Cite
@article{arxiv.1909.11027,
title = {Characterization of quasirandom permutations by a pattern sum},
author = {Timothy F. N. Chan and Daniel Kral and Jonathan A. Noel and Yanitsa Pehova and Maryam Sharifzadeh and Jan Volec},
journal= {arXiv preprint arXiv:1909.11027},
year = {2022}
}
Comments
Appendices 1-5 are contained in the ancillary pdf file available on arXiv for download