AISM 52, 415-425

## Joint distribution of rises and falls

### James C. Fu^{1} and W. Y. Wendy Lou^{2}

^{1}Department of Statistics, University of Manitoba,
Winnipeg, Manitoba, Canada R3T 2N2

^{2}Department of Biomathematical Sciences,
Mount Sinai School of Medicine, New York, NY 10029-6574, U.S.A.

(Received September 8, 1997; revised December 7, 1998)

Abstract.
The marginal distributions of the number of
rises and the number of falls have been used successfully
in various areas of statistics, especially in non-parametric
statistical inference. Carlitz (1972, *Duke Math. J.*, **39**, 268-269) showed that the generating function of the joint distribution for the numbers of
rises and falls satisfies certain complex combinatorial equations, and pointed out that he had been unable to derive the explicit formula for the joint distribution from these equations. After more than two decades, this latter problem remains unsolved. In this article, the joint distribution is obtained via the probabilistic method of finite Markov chain imbedding for random permutations. A numerical example is provided to illustrate the theoretical results and the corresponding computational procedures.

Key words and phrases:
Eulerian and Simon Newcomb numbers, finite Markov chain imbedding, transition probability matrix.

**Source**
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