Geometric and Functional Analysis

, Volume 23, Issue 2, pp 570–579 | Cite as

Quasirandom permutations are characterized by 4-point densities

  • Daniel Král’
  • Oleg Pikhurko
Open Access


For permutations \({\pi}\) and \({\tau}\) of lengths \({|\pi|\le|\tau|}\) , let \({t(\pi,\tau)}\) be the probability that the restriction of \({\tau}\) to a random \({|\pi|}\) -point set is (order) isomorphic to \({\pi}\) . We show that every sequence \({\{\tau_j\}}\) of permutations such that \({|\tau_j|\to\infty}\) and \({t(\pi,\tau_j)\to 1/4!}\) for every 4-point permutation \({\pi}\) is quasirandom (that is, \({t(\pi,\tau_j)\to 1/|\pi|!}\) for every \({\pi}\)). This answers a question posed by Graham.

Keywords and phrases

Permutations Quasirandomness Permutation limits Subpermutation density 


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© The Author(s) 2013

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  1. 1.Mathematics Institute, DIMAP and Department of Computer ScienceUniversity of WarwickCoventryUK
  2. 2.Faculty of Mathematics and Physics, Computer Science InstituteCharles UniversityPragueCzech Republic
  3. 3.Mathematics Institute and DIMAPUniversity of WarwickCoventryUK

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