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On the exact distributions of Eulerian and Simon Newcomb numbers associated with random permutations

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  • Fu, James C.
  • Lou, W. Y. Wendy
  • Wang, Yueh-Jir

Abstract

Eulerian and Simon Newcomb numbers are two of the most celebrated numbers associated with random permutations. Their distributions have been successfully used in various areas of statistics and applied probability. Conventionally, these distributions have been studied via combinatorial analysis. In this article, we provide a new, simple and unified probabilistic method based on the finite Markov chain imbedding technique to study the exact distributions of Eulerian and Simon Newcomb numbers. A new recursive equation which characterizes the Simon Newcomb numbers is obtained. We also show that many classical identities and recursive equations associated with Eulerian numbers are immediate consequences of our main result.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:stapro:v:42:y:1999:i:2:p:115-125
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    References listed on IDEAS

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    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. 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.
    3. M. Koutras & V. Alexandrou, 1995. "Runs, scans and URN model distributions: A unified Markov chain approach," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(4), pages 743-766, December.
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    Cited by:

    1. 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.
    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. 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. 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.
    6. 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.

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