IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v25y2023i2d10.1007_s11009-023-09987-0.html
   My bibliography  Save this article

On Distribution of the Number of Peaks and the Euler Numbers of Permutations

Author

Listed:
  • James C. Fu

    (University of Manitoba)

  • Wan-Chen Lee

    (Carleton University)

  • Hsing-Ming Chang

    (National Cheng Kung University)

Abstract

Using the language of runs and patterns, a peak in a sequence of integers can be interpreted as observing a fall (or descent) immediately after a rise (or ascent). In this paper, we obtain the exact distribution of the number of peaks in a permutation by using the nonhomogeneous finite Markov chain imbedding technique and an insertion procedure. As a byproduct, we also obtain the Euler numbers, which are a sequence of the number of alternating permutations. The method is extended to obtaining the joint distribution of the number of peaks and the number of falls. Several numerical examples are given to illustrate our theoretical results.

Suggested Citation

  • James C. Fu & Wan-Chen Lee & Hsing-Ming Chang, 2023. "On Distribution of the Number of Peaks and the Euler Numbers of Permutations," Methodology and Computing in Applied Probability, Springer, vol. 25(2), pages 1-13, June.
    • Handle: RePEc:spr:metcap:v:25:y:2023:i:2:d:10.1007_s11009-023-09987-0
    DOI: 10.1007/s11009-023-09987-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-023-09987-0
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11009-023-09987-0?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. 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.
    2. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Wang, Xiaoyue & Chen, Xi & Zhao, Xian & Ning, Ru, 2024. "Reliability analysis of self-healing systems equipped with multi-component protective devices operating in a shock environment," Reliability Engineering and System Safety, Elsevier, vol. 244(C).

    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, 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.
    2. Yu-Fei Hsieh & Tung-Lung Wu, 2013. "Recursive equations in finite Markov chain imbedding," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(3), pages 513-527, June.
    3. 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.
    4. 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.
    5. 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.
    6. 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.

    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:25:y:2023:i:2:d:10.1007_s11009-023-09987-0. 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.