IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v3y2001i1d10.1023_a1011414107588.html
   My bibliography  Save this article

Distribution of Increasing ℓ-sequences in a Random Permutation

Author

Listed:
  • Brad C. Johnson

    (University of Manitoba)

Abstract

This paper examines the distribution of the number, k, of increasing ℓ-sequences in a random permutation of $$\left\{ {1,...,n} \right\}$$ . A new solution is determined based on the compositions of n which requires, at most, $$k\left( {n - k - \ell } \right)$$ summands. This solution easily yields existing results for the special case $$\ell = 2$$ and provides an alternate form for the case $$\ell = 3$$ . The expected number of increasing ℓ-sequences in a random permutation is determined and it is shown that the limiting distribution is degenerate about 0 for $$\ell > 2$$ . An alternate algorithm to determine the exact distribution is presented, based on the partitions of n, which is easy to implement and efficient for small n. Applications in non-parametric statistics and graph theory are discussed.

Suggested Citation

  • Brad C. Johnson, 2001. "Distribution of Increasing ℓ-sequences in a Random Permutation," Methodology and Computing in Applied Probability, Springer, vol. 3(1), pages 35-49, March.
    • Handle: RePEc:spr:metcap:v:3:y:2001:i:1:d:10.1023_a:1011414107588
    DOI: 10.1023/A:1011414107588
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1023/A:1011414107588
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1023/A:1011414107588?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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. 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. 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.
    4. 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.
    5. Koutras, M. V. & Alexandrou, V. A., 1997. "Non-parametric randomness tests based on success runs of fixed length," Statistics & Probability Letters, Elsevier, vol. 32(4), pages 393-404, April.
    6. Tung-Lung Wu, 2013. "On Finite Markov Chain Imbedding and Its Applications," Methodology and Computing in Applied Probability, Springer, vol. 15(2), pages 453-465, June.
    7. 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.

    More about this item

    Keywords

    permutations; increasing ℓ-sequences;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:metcap:v:3:y:2001:i:1:d:10.1023_a:1011414107588. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.