Asymptotic normality of the generalized Eulerian numbers (Q2715953)
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scientific article; zbMATH DE number 1600926
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Asymptotic normality of the generalized Eulerian numbers |
scientific article; zbMATH DE number 1600926 |
Statements
30 May 2001
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Eulerian number
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permutation
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normal distribution
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Asymptotic normality of the generalized Eulerian numbers (English)
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A permutation \(\pi \in S_n\) of \(\{1, 2, \ldots , n\}\) has an \(m\)-rise if \(\pi (i+1)-\pi (i)\geq m\) for some \(i\). This interesting article considers the random variable \(X\) on \(S_n\) (all \(n!\) permutations are equiprobable) defined by \(X(\pi)=\) the number of \(m\)-rises of \(\pi \). It is proven that for \(n\rightarrow \infty \) and \(m=o(n)\) the distribution function of \(X\) converges weakly to a normal distribution.
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