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

✍ Scribed by James C. Fu; W.Y.Wendy Lou; Yueh-Jir Wang

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 111 KB
• Volume: 42
• Category: Article
• ISSN: 0167-7152
• DOI: 10.1016/s0167-7152(98)00176-x

✦ Synopsis

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 uniÿed probabilistic method based on the ÿnite 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.