Patterns in Random Permutations


Every k entries in a permutation can have one of k! different relative orders, called patterns. How many times does each pattern occur in a large random permutation of size n?

The distribution of this k!-dimensional vector of pattern densities was studied by Janson, Nakamura, and Zeilberger (2015). Their analysis showed that some component of this vector is asymptotically multi-normal of order \(1/\sqrt n \), while the orthogonal component is smaller.

Using representations of the symmetric group, and the theory of U-statistics, we refine the analysis of this distribution. We show that it decomposes into k asymptotically uncorrelated components of different orders in n, that correspond to Sk-representations.

Some combinations of pattern densities that arise in this decomposition have interpretations as practical nonparametric statistical tests.

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  1. [1]

    M. H. Albert, M. D. Atkinson, C. C. Handley, D. A. Holton and W. Stromquist: On packing densities of permutations, Electron. J. Combin 9, (2002).

  2. [2]

    W. Bergsma and A. Dassios: A consistent test of independence based on a sign covariance related to Kendall’s tau, Bernoulli 20 (2014), 1006–1028.

    MathSciNet  Article  Google Scholar 

  3. [3]

    W. Bergsma: Nonparametric testing of conditional independence by means of the partial copula, 2010.

  4. [4]

    A. Burstein and P. Hästö: Packing sets of patterns, European Journal of Combinatorics 31 (2010), 241–253.

    MathSciNet  Article  Google Scholar 

  5. [5]

    J. Balogh, P. Hu, B. Lidický, O. Pikhurko, B. Udvari and J. Volec: Minimum number of monotone subsequences of length 4 in permutations, Combinatorics, Probability and Computing 24 (2015), 658–679.

    MathSciNet  Article  Google Scholar 

  6. [6]

    J. R. Blum, J. C. Kiefer and M. Rosenblatt: Distribution free tests of independence based on the sample distribution function, The Annals of Mathematical Statistics, (1961), 485–498.

  7. [7]

    M. Bóna: The copies of any permutation pattern are asymptotically normal, arXiv:0712.2792, 2007.

  8. [8]

    M. Bóna: On three different notions of monotone subsequences, Permutation Patterns 376 (2010), 89–114.

    MathSciNet  Article  Google Scholar 

  9. [9]

    M. Bóna: Combinatorics of Permutations, CRC Press, 2012.

  10. [10]

    J. N. Cooper: Quasirandom permutations, Journal of Combinatorial Theory, Series A 106 (2004), 123–143.

    MathSciNet  Article  Google Scholar 

  11. [11]

    J. N. Cooper: A permutation regularity lemma, The Electronic Journal of Combinatorics 13 (2006), 22.

    MathSciNet  Article  Google Scholar 

  12. [12]

    P. Diaconis: Group representations in probability and statistics, Lecture Notes — Monograph Series 11 (1988), 1–192.

    MathSciNet  MATH  Google Scholar 

  13. [13]

    A. B. Dieker and F. V. Saliola: Spectral analysis of random-to-random Markov chains, Advances in Mathematics 323 (2018), 427–485.

    MathSciNet  Article  Google Scholar 

  14. [14]

    C. Even-Zohar: GitHub repository patterns, 2018.

  15. [15]

    W. Fulton and J. Harris: Representation Theory: A First Course, volume 129. Springer Science & Business Media, 1991.

  16. [16]

    N. I. Fisher and A. J. Lee: Nonparametric measures of angular-angular association, Biometrika (1982), 315–321.

  17. [17]

    J. Fulman: Stein’s method and non-reversible Markov chains, in: Stein’s Method, pages 66–74. Institute of Mathematical Statistics, 2004.

  18. [18]

    R. Glebov, A. Grzesik, T. Klimošová and D. Král’: Finitely forcible graphons and permutons, Journal of Combinatorial Theory, Series B 110 (2015), 112–135.

    MathSciNet  Article  Google Scholar 

  19. [19]

    P. A. Hästö: The packing density of other layered permutations, Journal of Combinatorics 9 (2002), 1.

    MathSciNet  Article  Google Scholar 

  20. [20]

    C. Hoppen, Y. Kohayakawa, C. G. Moreira, B. Ráth and R. M. Sampaio: Limits of permutation sequences, Journal of Combinatorial Theory, Series B 103 (2013), 93–113.

    MathSciNet  Article  Google Scholar 

  21. [21]

    C. Hoppen, Y. Kohayakawa, C. G. Moreira and R. M. Sampaio: Testing permutation properties through subpermutations, Theoretical Computer Science 412 (2011), 3555–3567.

    MathSciNet  Article  Google Scholar 

  22. [22]

    W. Hoeffding: A class of statistics with asymptotically normal distribution, The Annals of Mathematical Statistics 19 (1948), 293–325.

    MathSciNet  Article  Google Scholar 

  23. [23]

    W. Hoeffding: A non-parametric test of independence, The Annals of Mathematical Statistics (1948), 546–557.

  24. [24]

    L. Hofer: A central limit theorem for vincular permutation patterns, arXiv:1704.00650, 2017.

  25. [25]

    S. Janson: Gaussian Hilbert Spaces, volume 129, Cambridge University Press, 1997.

  26. [26]

    S. Janson, B. Nakamura and D. Zeilberger: On the asymptotic statistics of the number of occurrences of multiple permutation patterns, Journal of Combinatorics 6 (2015), 117–143.

    MathSciNet  Article  Google Scholar 

  27. [27]

    V. S. Korolyuk and Y. V. Borovskich: Theory of U-statistics, volume 273, Springer Science & Business Media, 2013.

  28. [28]

    M. G. Kendall: A new measure of rank correlation, Biometrika 30 (1938), 81–93.

    Article  Google Scholar 

  29. [29]

    S. Kitaev: Patterns in Permutations and Words, Springer Science & Business Media, 2011.

  30. [30]

    T. Klimošová and D. Král’: Hereditary properties of permutations are strongly testable, in: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, 1164–1173, Society for Industrial and Applied Mathematics, 2014.

  31. [31]

    R. Kenyon, D. Král’, C. Radin and P. Winkler: Permutations with fixed pattern densities, arXiv:1506.02340, 2015.

  32. [32]

    D. Král’ and O. Pikhurko: Quasirandom permutations are characterized by 4-point densities, Geometric and Functional Analysis 23 (2013), 570–579.

    MathSciNet  Article  Google Scholar 

  33. [33]

    J. Lee: U-Statistics: Theory and Practice, volume 110, Marcel Dekker, Inc., 1990.

  34. [34]

    A. Marcus and G. Tardos: Excluded permutation matrices and the Stanley-Wilf conjecture, Journal of Combinatorial Theory, Series A 107 (2004), 153–160.

    MathSciNet  Article  Google Scholar 

  35. [35]

    A. L. Price: Packing densities of layered *patterns, 1997, Dissertations available from ProQuest. AAI9727276.

  36. [36]

    C. B. Presutti and W. Stromquist: Packing rates of measures and a conjecture for the packing density of 2413, Permutation Patterns 376 (2010), 287–316.

    MathSciNet  Article  Google Scholar 

  37. [37]

    Sage Developers: SageMath, the Sage Mathematics Software System, Versions 7.4 and 8.1, 2018,

  38. [38]

    C. Spearman: The proof and measurement of association between two things, The American Journal of Psychology 15 (1904), 72–101.

    Article  Google Scholar 

  39. [39]

    J. Sliacan and W. Stromquist: Improving bounds on packing densities of 4-point permutations, arXiv:1704.02959, 2017.

  40. [40]

    D. Zeilberger: Doron Gepner’s statistics on words in {1, 2, 3} is (most probably) asymptotically logistic, arXiv:1604.00663, 2016.

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The author thanks Eric Babson, Svante Janson, Nati Linial, and Doron Zeilberger for insightful discussions, and the anonymous referees for their helpful comments.

Some of the computational work was carried out with the facilities of the School of Computer Science and Engineering at HUJI, supported by ERC 339096.

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Correspondence to Chaim Even-Zohar.

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Even-Zohar, C. Patterns in Random Permutations. Combinatorica 40, 775–804 (2020).

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Mathematics Subject Classification (2010)

  • 05A05