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EXACT AND LIMITING DISTRIBUTIONS OF THE NUMBER

OF SUCCESSIONS IN A RANDOM PERMUTATION

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JAMES C. FU

*Institute of Applied Mathematics, National Donghwa
University,
Hualien, Taiwan, R.O.C.*
(Received April 14, 1994; revised November 7, 1994)

**Abstract.**
Traditionally the distributions of the
number of patterns and successions in a random permutation
of *n* integers 1, 2, ..., and *n* 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 *lambda* = 1. Furthermore, a
direct application of the main result, it also yields the
waiting time distribution of a succession.

*Key words and phrases*:
Permutation,
succession, Markov chain imbedding, transition
probabilities, Poisson convergence, waiting time.

**Source**
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