Abstract
Traditionally the distributions of the number of patterns and successions in a random permutation ofn integers 1,2, ..., andn were studied by combinatorial analysis. In this short article, a simple way based on finite Markov chain imbedding technique is used to obtain the exact distribution of successions on a permutation. This approach also gives a direct proof that the limiting distribution of successions is a Poisson distribution with parameter λ=1. Furthermore, a direct application of the main result, it also yields the waiting time distribution of a succession.
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This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant NSERC A-9216, and National Science Council of Republic of China under Grant 85-2121-M-259-003.
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Fu, J.C. Exact and limiting distributions of the number of successions in a random permutation. Ann Inst Stat Math 47, 435–446 (1995). https://doi.org/10.1007/BF00773393
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DOI: https://doi.org/10.1007/BF00773393