On the maximum of \(r\)-Stirling numbers
\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.
scientific article; zbMATH DE number 5355923
scientific article