Patterns in Random Permutations Avoiding Some Other Patterns (Keynote Speakers)

Author Svante Janson

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Svante Janson
  • Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden

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Svante Janson. Patterns in Random Permutations Avoiding Some Other Patterns (Keynote Speakers). In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 6:1-6:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


Consider a random permutation drawn from the set of permutations of length n that avoid a given set of one or several patterns of length 3. We show that the number of occurrences of another pattern has a limit distribution, after suitable scaling. In several cases, the limit is normal, as it is in the case of unrestricted random permutations; in other cases the limit is a non-normal distribution, depending on the studied pattern. In the case when a single pattern of length 3 is forbidden, the limit distributions can be expressed in terms of a Brownian excursion. The analysis is made case by case; unfortunately, no general method is known, and no general pattern emerges from the results.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Permutations and combinations
  • Mathematics of computing → Probabilistic representations
  • Random permutations
  • patterns
  • forbidden patterns
  • limit in distribution
  • U-statistics


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  1. Frédérique Bassino, Mathilde Bouvel, Valentin Féray, Lucas Gerin, Mickaël Maazoun, and Adeline Pierrot. Universal limits of substitution-closed permutation classes. Preprint,, 2017.
  2. Frédérique Bassino, Mathilde Bouvel, Valentin Féray, Lucas Gerin, and Adeline Pierrot. The Brownian limit of separable permutations. Preprint,, 2016.
  3. Miklós Bóna. Combinatorics of Permutations. Chapman &Hall/CRC, Boca Raton, FL, 2004. Google Scholar
  4. Miklós Bóna. The copies of any permutation pattern are asymptotically normal. Preprint,, 2007.
  5. Miklós Bóna. On three different notions of monotone subsequences. In Permutation patterns, volume 376 of London Math. Soc. Lecture Note Ser., pages 89-114. Cambridge Univ. Press, Cambridge, 2010. Google Scholar
  6. Svante Janson. Gaussian Hilbert Spaces. Cambridge University Press, Cambridge, 1997. Google Scholar
  7. Svante Janson. Brownian excursion area, Wright’s constants in graph enumeration, and other Brownian areas. Probab. Surv., 4:80-145, 2007. Google Scholar
  8. Svante Janson. Patterns in random permutations avoiding the pattern 132. Combin. Probab. Comput., 26(1):24-51, 2017. Google Scholar
  9. Svante Janson. Patterns in random permutations avoiding the pattern 321. Preprint,, 2017.
  10. Svante Janson. Patterns in random permutations avoiding some sets of multiple patterns. Preprint,, 2018.
  11. Svante Janson. Renewal theory for asymmetric U-statistics. Preprint,, 2018.
  12. Svante Janson, Brian Nakamura, and Doron Zeilberger. On the asymptotic statistics of the number of occurrences of multiple permutation patterns. J. Comb., 6(1-2):117-143, 2015. Google Scholar
  13. Donald E. Knuth. The Art of Computer Programming. Vol. 1. Addison-Wesley, Reading, MA, third edition, 1997. Google Scholar
  14. Rodica Simion and Frank W. Schmidt. Restricted permutations. European J. Combin., 6(4):383-406, 1985. Google Scholar
  15. Richard P. Stanley. Enumerative combinatorics. Vol. 2. Cambridge University Press, Cambridge, 1999. Google Scholar
  16. Robert Tarjan. Sorting using networks of queues and stacks. J. Assoc. Comput. Mach., 19:341-346, 1972. Google Scholar
  17. Julian West. Generating trees and the Catalan and Schröder numbers. Discrete Math., 146(1-3):247-262, 1995. Google Scholar