The Annals of Probability

The Brownian limit of separable permutations

Frédérique Bassino, Mathilde Bouvel, Valentin Féray, Lucas Gerin, and Adeline Pierrot

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We study uniform random permutations in an important class of pattern-avoiding permutations: the separable permutations. We describe the asymptotics of the number of occurrences of any fixed given pattern in such a random permutation in terms of the Brownian excursion. In the recent terminology of permutons, our work can be interpreted as the convergence of uniform random separable permutations towards a “Brownian separable permuton”.

Article information

Ann. Probab., Volume 46, Number 4 (2018), 2134-2189.

Received: April 2016
Revised: February 2017
First available in Project Euclid: 13 June 2018

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Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 05A05: Permutations, words, matrices

Permutation patterns Brownian excursion permutons


Bassino, Frédérique; Bouvel, Mathilde; Féray, Valentin; Gerin, Lucas; Pierrot, Adeline. The Brownian limit of separable permutations. Ann. Probab. 46 (2018), no. 4, 2134--2189. doi:10.1214/17-AOP1223.

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  • [1] Albert, M., Homberger, C. and Pantone, J. (2015). Equipopularity classes in the separable permutations. Electron. J. Combin. 22 Paper 2.2.
  • [2] Albert, M. H. and Atkinson, M. D. (2005). Simple permutations and pattern restricted permutations. Discrete Math. 300 1–15. DOI:10.1016/j.disc.2005.06.016.
  • [3] Aldous, D. (1993). The continuum random tree. III. Ann. Probab. 21 248–289.
  • [4] Atapour, M. and Madras, N. (2014). Large deviations and ratio limit theorems for pattern-avoiding permutations. Combin. Probab. Comput. 23 160–200.
  • [5] Avis, D. and Newborn, M. (1981). On pop-stacks in series. Util. Math. 19 129–140.
  • [6] Bassino, F., Bouvel, M., Féray, V., Gerin, L., Maazoun, M. and Pierrot, A. Universal limits of substitution-closed permutation classes. Available at arXiv:1706.08333.
  • [7] Bevan, D. (2015). On the growth of permutation classes Ph.D. thesis, Open University. Available at arXiv:1506.06688.
  • [8] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York. DOI:10.1002/9780470316962.
  • [9] Bogachev, V. I. (2007). Measure Theory, Vol. 2. Springer, Berlin.
  • [10] Bóna, M. (2010). The absence of a pattern and the occurrences of another. Discrete Math. Theor. Comput. Sci. 12 89–102.
  • [11] Bóna, M. (2012). Surprising symmetries in objects counted by Catalan numbers. Electron. J. Combin. 19 Paper 62.
  • [12] Bóna, M. (2012). Combinatorics of Permutations, 2nd ed. Chapman-Hall and CRC Press, London.
  • [13] Bose, P., Buss, J. F. and Lubiw, A. (1998). Pattern matching for permutations. Inform. Process. Lett. 65 277–283. DOI:10.1016/S0020-0190(97)00209-3.
  • [14] Cheng, S.-E., Eu, S.-P. and Fu, T.-S. (2007). Area of Catalan paths on a checkerboard. European J. Combin. 28 1331–1344.
  • [15] The Sage Developers (2016). Sage mathematics software (Version 7.1). Availabel at
  • [16] Dokos, T. and Pak, I. (2014). The expected shape of random doubly alternating Baxter permutations. Online J. Anal. Comb. 9 Article 5.
  • [17] Freedman, D. (2012). Brownian Motion and Diffusion, 3rd ed. Springer, Berlin.
  • [18] Ghys, E. A Singular Mathematical Promenade. ENS Éditions, Lyon.
  • [19] Glebov, R., Grzesik, A., Klimosová, T. and Král’, D. (2015). Finitely forcible graphons and permutons. J. Combin. Theory Ser. B 110 112–135.
  • [20] Hoffman, C., Rizzolo, D. and Slivken, E. (2016). Pattern avoiding permutations and Brownian excursion part II: Fixed points. Probab. Theory Related Fields 169 377–424.
  • [21] Hoffman, C., Rizzolo, D. and Slivken, E. (2017). Pattern avoiding permutations and Brownian excursion part I: Shapes and fluctuations. Random Structures Algorithms 50 394–419.
  • [22] Homberger, C. (2012). Expected patterns in permutation classes. Electron. J. Combin. 19 Paper 43.
  • [23] Hoppen, C., Kohayakawa, Y., Moreira, C. G., Rath, B. and Sampaio, R. M. (2013). Limits of permutation sequences. J. Combin. Theory Ser. B 103 93–113.
  • [24] Janson, S. (2017). Patterns in random permutations avoiding the pattern $132$. Combin. Probab. Comput. 26 24–51.
  • [25] Janson, S., Nakamura, B. and Zeilberger, D. (2015). On the asymptotic statistics of the number of occurrences of multiple permutation patterns. J. Comb. 6 117–143.
  • [26] Kallenberg, O. (2006). Foundations of Modern Probability, Springer, New York.
  • [27] Kenyon, R., Král’, D., Radin, Ch. and Winkler, P. (2015). Permutations with fixed pattern densities (previous title: A variational principle for permutations). Available at arXiv:1506.02340.
  • [28] Kitaev, S. (2011). Patterns in Permutations and Words. Springer, Berlin.
  • [29] Kortchemski, I. (2012). Invariance principles for Galton–Watson trees conditioned on the number of leaves. Stochastic Process. Appl. 122 3126–3172. DOI:10.1016/
  • [30] Le Gall, J.-F. (2005). Random trees and applications. Probab. Surv. 2 245–311.
  • [31] Maazoun, M. (2017). On the Brownian separable permuton. In preparation.
  • [32] Madras, N. and Liu, H. (2010). Random pattern-avoiding permutations. In Algorithmic, Probability and Combinatorics. Contemp. Math. 520 173–194. Amer. Math. Soc., Providence, RI.
  • [33] Madras, N. and Pehlivan, L. (2016). Structure of random 312-avoiding permutations. Random Structures Algorithms 49 599–631. DOI:10.1002/rsa.20601.
  • [34] Miner, S. and Pak, I. (2014). The shape of random pattern-avoiding permutations. Adv. in Appl. Math. 55 86–130.
  • [35] Mörters, P. and Peres, Y. (2010). Brownian Motion. Cambridge Univ. Press, Cambridge.
  • [36] Pitman, J. (2006). Combinatorial Stochastic Processes, 2002 Saint-Flour Lecture Notes. Lecture Notes in Math. 1875. Springer, Berlin.
  • [37] Pitman, J. and Rizzolo, D. (2015). Schröder’s problems and scaling limits of random trees. Trans. Amer. Math. Soc. 367 6943–6969. DOI:10.1090/S0002-9947-2015-06254-0.
  • [38] Rudolph, K. (2013). Pattern popularity in 132-avoiding permutations. Electron. J. Combin. 20 Paper 8.
  • [39] Shapiro, L. and Stephens, A. B. (1991). Bootstrap percolation, the Schröder numbers, and the $n$-kings problem. SIAM J. Discrete Math. 4 275–280.
  • [40] Vatter, V. (2015). Permutation classes. In Handbook of Enumerative Combinatorics 753–833. CRC Press, Boca Raton, FL.
  • [41] Westenberg, M. A., Roerdink, J. B. T. M., Kuipers, O. P. and van Hijum, S. A. F. T. (2010). Random graphs and complex networks. Lecture notes. Available at
  • [42] Wikipedia. Enumerations of specific permutation classes. Available at Accessed on Jan. 5th, 2016.