On the maximum of \(r\)-Stirling numbers
type: Item
instance of: scholarly article
title: On the maximum of \(r\)-Stirling numbers
zbMATH Open document ID: 1165.11023
published in: Advances in Applied Mathematics
publication date: 2008-10-22T00:00:00Z
review text: \textit{A. Z. Broder} [Discrete Math. 49, 241--259 (1984; Zbl 0535.05006)] extensively studied \(r\)-Stirling numbers, which seem to have been introduced by Carlitz. By definition, the \(r\)-Stirling number of the first kind, \({n \brack m}_r\) counts the number of permutations of \(\{1,2,\ldots,n\}\) with \(m\) cycles, where the numbers \(1,2,\ldots,r\) belong to distinct cycles; and the \(r\)-Stirling number of the second kind, \({n \brace m}_r\) counts the number of partitions of \(\{1,2,\ldots,n\}\) with \(m\) non-empty classes, in which the numbers \(1,2,\ldots,r\) belong to distinct classes. The paper under review finds good approximation for the values of \(m\) that maximizes the \(r\)-Stirling number of the first or second kind, while \(n\) and \(r\) are fixed. Asymptotic normality of the \(r\)-Stirling numbers of the first and second kind are shown, after normalization, around their maximum places.
reviewed by: László A. Székely
Mathematics Subject Classification ID: 11B73
zbMATH DE Number: 5355923
zbMATH Keywords: Stirling numbers
zbMATH Keywords: \(r\)-Stirling numbers
zbMATH Keywords: unimodality
zbMATH Keywords: log-concavity
zbMATH Keywords: asymptotic normality
author: István Mező
MaRDI profile type: Publication
full work available at URL: j.aam.2007.11.002
OpenAlex ID: W2104319487
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DOI: 10.1016/J.AAM.2007.11.002
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description: scientific article; zbMATH DE number 5355923