A probabilistic approach to the descent statistic
type: Item
instance of: scholarly article
title: A probabilistic approach to the descent statistic
zbMATH Open document ID: 1005.05005
author: R. Smith
published in: Journal of Combinatorial Theory. Series A
publication date: 2002-06-26T00:00:00Z
review text: Given a permutation \(\sigma= \sigma_1\cdots \sigma_{n+1}\) of \(\{1,\dots, n+1\}\), the decent word \(u= u_1\cdots u_n\) is the word in variables \(a\) and \(b\) with \(u_i= a\) if \(\sigma_i> \sigma_{i+1}\) and \(b\) otherwise. For an \(ab\)-word \(u\), the decent statistic is the number of permutations having \(u\) as decent word. The authors present a probabilistic approach to studying the decent based upon a two-variable probability density deriving quadratic inequalities for the descent statistic. Using Fourier series, they give exact expressions for the Euler numbers and the alternating \(r\)-signed permutations. They also obtain a probabilistic interpretation of the sin function.
Mathematics Subject Classification ID: 05A15
zbMATH DE Number: 1760673
zbMATH Keywords: permutation
zbMATH Keywords: decent word
zbMATH Keywords: Euler numbers
reviewed by: Jack E. Graver
MaRDI profile type: Publication
full work available at URL: jcta.2001.3233
OpenAlex ID: W2150654349
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DOI: 10.1006/JCTA.2001.3233
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description: scientific article; zbMATH DE number 1760673