Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.8

Mode and Edgeworth Expansion for the Ewens Distribution and the Stirling Numbers

Zakhar Kabluchko
Institut für Mathematische Statistik
Universität Münster
Orléans–Ring 10
48149 Münster

Alexander Marynych
Faculty of Cybernetics
Taras Shevchenko National University of Kyiv
01601 Kyiv

Henning Sulzbach
School of Computer Science
McGill University
3480 University Street
Montréal, QC H3A 0E9


We provide asymptotic expansions for the Stirling numbers of the first kind and, more generally, the Ewens (or Karamata-Stirling) distribution. Based on these expansions, we obtain some new results on the asymptotic properties of the mode and the maximum of the Stirling numbers and the Ewens distribution. For arbitrary $\theta>0$ and for all sufficiently large $n\in\mathbb{N} $, the unique maximum of the Ewens probability mass function

\begin{displaymath}\mathbb L_n(k) = \frac{\theta^k}{\theta(\theta+1)\cdots(\theta+n-1)} \genfrac{[}{]}{0pt}{}{n}{k}, \quad k=1,\ldots,n,

is attained at $k= \lfloor a_n \rfloor$ or $\lceil a_n \rceil$, where $a_n = \theta \log n - \theta \Gamma'(\theta)/\Gamma(\theta) -1/2$. We prove that the mode is the nearest integer to an for a set of n's of asymptotic density 1, yet this formula is not true for infinitely many n's.

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(Concerned with sequences A008275 A048994 A132393.)

Received September 30 2016. Revised version received November 21 2016. Published in Journal of Integer Sequences, November 25 2016.

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