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On the Distribution of the Number of Occurrences of an Order-Preserving Pattern of Length Three in a Random Permutation

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  • James C. Fu

    (University of Manitoba)

Abstract

Recently, a considerable number of papers in computer science and mathematics examined the number of permutations containing exactly s occurrences of a prescribed order-preserving pattern (or forbidden pattern). It is well known that, mathematically, this is an NP-hard problem. Even in the simple case where the length of an order-preserving pattern is three, the number of permutations of size n containing s (s ≥ 3) order-preserving patterns remains unknown (see Fulmek Adv Appl Math 30(4):607–632, 2003). This manuscript provides a probabilistic approach to enumerate the number of permutations that contain exactly s occurrences of an order-preserving pattern of length three. The method is based on the insertion procedure of the finite Markov chain imbedding technique. Numerical results are provided to illustrate the theoretical results.

Suggested Citation

  • James C. Fu, 2012. "On the Distribution of the Number of Occurrences of an Order-Preserving Pattern of Length Three in a Random Permutation," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 831-842, September.
    • Handle: RePEc:spr:metcap:v:14:y:2012:i:3:d:10.1007_s11009-012-9279-6
    DOI: 10.1007/s11009-012-9279-6
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    References listed on IDEAS

    as
    1. James Fu, 1995. "Exact and limiting distributions of the number of successions in a random permutation," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(3), pages 435-446, September.
    2. Johnson, Brad C., 2002. "The distribution of increasing 2-sequences in random permutations of arbitrary multi-sets," Statistics & Probability Letters, Elsevier, vol. 59(1), pages 67-74, August.
    3. Harris, Bernard & Park, C. J., 1994. "A generalization of the Eulerian numbers with a probabilistic application," Statistics & Probability Letters, Elsevier, vol. 20(1), pages 37-47, May.
    4. Fu, James C. & Lou, W. Y. Wendy & Wang, Yueh-Jir, 1999. "On the exact distributions of Eulerian and Simon Newcomb numbers associated with random permutations," Statistics & Probability Letters, Elsevier, vol. 42(2), pages 115-125, April.
    5. Johnson, Brad C. & Fu, James C., 2000. "The distribution of increasing l-sequences in random permutations: a Markov chain approach," Statistics & Probability Letters, Elsevier, vol. 49(4), pages 337-344, October.
    6. James Fu & W.Y. Lou, 2000. "Joint Distribution of Rises and Falls," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(3), pages 415-425, September.
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    Cited by:

    1. James C. Fu & Yu-Fei Hsieh, 2015. "On the Distribution of the Length of the Longest Increasing Subsequence in a Random Permutation," Methodology and Computing in Applied Probability, Springer, vol. 17(2), pages 489-496, June.
    2. Anant P. Godbole & Martha Liendo, 2016. "Waiting Time Distribution for the Emergence of Superpatterns," Methodology and Computing in Applied Probability, Springer, vol. 18(2), pages 517-528, June.

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