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On the exact distributions of Eulerian and Simon Newcomb numbers associated with random permutations. (English) Zbl 1057.62503
Summary: 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. 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.

MSC:
62E15 Exact distribution theory in statistics
11B68 Bernoulli and Euler numbers and polynomials
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