\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{graphicx} \usepackage{amscd} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \definecolor{webgreen}{rgb}{0,.5,0} \definecolor{webbrown}{rgb}{.6,0,0} \usepackage{color} \usepackage{fullpage} \usepackage{float} \usepackage{psfig} \usepackage{graphics,amsmath,amssymb} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{latexsym} \usepackage{epsf} \usepackage{authblk} \usepackage{mathrsfs} \usepackage{bbm} \usepackage[numbers]{natbib} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.1in} \setlength{\textheight}{8.4in} \newcommand{\seqnum}[1]{\href{http://oeis.org/#1}{\underline{#1}}} \DeclareMathOperator*{\argmin}{arg\,min} \DeclareMathOperator*{\dist}{dist} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \begin{center} \vskip 1cm{\LARGE\bf Mode and Edgeworth Expansion for the \\ \vskip .1in Ewens Distribution and the Stirling Numbers } \vskip 1cm \large Zakhar Kabluchko\\ Institut f\"ur Mathematische Statistik\\ Universit\"at M\"unster\\ Or\-l\'e\-ans--Ring 10\\ 48149 M\"unster \\ Germany\\ \href{mailto:zakhar.kabluchko@uni-muenster.de}{\tt zakhar.kabluchko@uni-muenster.de}\\ \ \\ Alexander Marynych\\ Faculty of Cybernetics\\ Taras Shevchenko National University of Kyiv\\ 01601 Kyiv\\ Ukraine\\ \href{mailto:marynych@unicyb.kiev.ua}{\tt marynych@unicyb.kiev.ua}\\ \ \\ Henning Sulzbach\\ School of Computer Science\\ McGill University\\ 3480 University Street \\ Montr\'eal, QC H3A 0E9 \\ Canada\\ \href{mailto:henning.sulzbach@gmail.com}{\tt henning.sulzbach@gmail.com} \end{center} \vskip .2 in \renewcommand{\P}{\mathbb{P}} \newcommand{\bL}{\mathbb{L}} \newcommand{\Herm}{\mathrm{He}} \newcommand{\fD}{\mathscr{D}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\C}{\mathbb{C}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\ind}{\mathbbm{1}} \newcommand{\dd}{{\rm d}} \newcommand{\eee}{e} \newcommand{\nint}{\mathop{\mathrm{nint}}\nolimits} \newcommand{\stirling}[2]{\genfrac{[}{]}{0pt}{}{#1}{#2}} \newcommand{\stirlingb}[2]{\genfrac{\{}{\}}{0pt}{}{#1}{#2}} \newcommand{\eps}{\varepsilon} \newcommand{\eqdistr}{\stackrel{d}{=}} \newcommand{\todistr}{\overset{d}{\underset{n\to\infty}\longrightarrow}} \newcommand{\ton}{\overset{}{\underset{n\to\infty}\longrightarrow}} \vfill\eject \begin{abstract} 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 $n\in\N$, the unique maximum of the Ewens probability mass function $$ \mathbb L_n(k) = \frac{\theta^k}{\theta(\theta+1)\cdots(\theta+n-1)} \stirling {n}{k}, \quad k=1,\ldots,n, $$ 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 $a_n$ for a set of $n$'s of asymptotic density $1$, yet this formula is not true for infinitely many $n$'s. \end{abstract} \section{Introduction and statement of results} \subsection{Introduction} The (unsigned) \textit{Stirling numbers} of the first kind $\stirling{n}{k}$ are defined, for $n\in\N$ and $1\leq k\leq n$, by the formula \begin{equation}\label{eq:stirling_def} x^{(n)} := x(x+1)\cdots (x+n-1) = \sum_{k=1}^n \stirling{n}{k} x^k,\quad x\in\R. \end{equation} For $n\in\N$, a random variable $K_n(\theta)$ is said to have {the} {\it Ewens distribution} with parameter $\theta>0$ if its probability mass function is given by the formula $$ \P(K_n(\theta)=k)=\frac{\theta^k}{\theta^{(n)}}\stirling{n}{k},\quad k=1,\ldots,n. $$ Bingham \cite{bingham_tauberian} called this distribution the \textit{Karamata-Stirling law}. One can interpret $K_n(\theta)$ as the number of blocks in a random partition of $\{1,\ldots,n\}$ distributed according to the Ewens sampling formula, or, equivalently, the number of different alleles in the infinite alleles model, the number of tables in a Chinese restaurant process, or the number of colors in the Hoppe urn. The Ewens sampling formula plays an important role in population genetics \cite{ewens_tavare}, \cite[Section~1.3]{durrett_DNA}. There is a distributional representation of $K_n(\theta)$ as a sum of independent random variables \begin{equation*} K_n(\theta) \eqdistr \xi_1+\cdots+\xi_n, \text{ where } \xi_i\sim \text{Bern}(\theta/(\theta + i-1)) \end{equation*} and $\text{Bern}(p)$ denotes the Bernoulli distribution with parameter $p$. In the special case $\theta=1$, {classical results going back at least to Feller \cite{feller1945} and R\'enyi \cite{renyi} state that} the random variable $K_n(1)$ has the same distribution as the number of cycles in a uniformly chosen random permutation of $n$ objects, or the number of records in a sample of $n$ i.i.d.\ variables from a continuous distribution. It follows easily from Lindeberg's theorem that $K_n(\theta)$ satisfies a central limit theorem of the form $$ \frac{K_n(\theta) - \theta \log n}{\sqrt{\theta \log n}} \todistr \text{N}(0,1) $$ known as Goncharov's CLT in the case $\theta=1$. Asymptotic expansions, as $n\to\infty$, of the Stirling numbers $\stirling {n}{k}$ in various regions of $k$ were provided in numerous works \cite{hwang_diss,hwang_stirling,louchard_first_kind,moser_wyman,temme,wilf}. Most notably, \citet[Theorem~2]{hwang_stirling} (and Theorem~14 on page 108 of his dissertation \cite{hwang_diss} for a more general result) gave an asymptotic expansion valid uniformly in the domain $2\leq k \leq \eta \log n$, for any fixed $\eta>0$. Louchard~\cite[Theorem~2.1]{louchard_first_kind} computed three non-trivial terms of the asymptotic expansion in the central regime $k= \log n + O(\sqrt {\log n})$ which is similar to the classical Edgeworth expansion in the central limit theorem. In this short note we start by deriving a full Edgeworth expansion, as $n\to\infty$, for the sequence of probability mass functions $k\mapsto \P(K_n(\theta)=k)$ which is uniform both in $\theta\in [1/\eta, \eta]$ (where $\eta>1$) and in $k\in \{1,\ldots,n\}$; see Theorem~\ref{theo:stirling_edgeworth}. Our result is an application of the general Edgeworth expansion for deterministic or random profiles which the authors \cite{Kabluchko+Marynych+Sulzbach:2016} recently obtained. Using this asymptotic expansion we derive some new results on the mode and the maximum of the Ewens distribution. In the case $\theta=1$ the mode can be interpreted as the most probable number of cycles in a random permutation of $n$ objects. It was investigated in the works of \citet{hammersley_restricted, hammersley} and Erd\H{o}s \cite{erdoes_on_hammersley}. Our results on the mode and the maximum will be stated in Theorems~\ref{theo:stirling_mode_maximum} and~\ref{theo:stirling_mode_further} below. \subsection{Asymptotic expansion of the Ewens distribution} Before stating our main result we need to recall some notions. The (complete) \textit{Bell polynomials} $B_j(z_1,\ldots,z_j)$ are defined by the formal identity \begin{equation*} \exp\left( \sum_{j=1}^{\infty} \frac{x^j}{j!} z_j \right) = \sum_{j=0}^{\infty} \frac{x^j}{j!} B_j(z_1,\ldots,z_j). \end{equation*} Therefore $B_0=1$ and, for $j\in\N$, \begin{equation}\label{eq:bell_poly_def} B_j(z_1,\ldots,z_j) = \sum{}^{'}\frac{j!}{i_1!\cdots i_j!} \left(\frac{z_1}{1!}\right)^{i_1}\cdots \left(\frac{z_j}{j!}\right)^{i_j}, \end{equation} where the sum $\sum{}^{'}$ is taken over all $i_1,\ldots,i_j\in\N_0$ satisfying $1i_1+2i_2+\cdots +j i_j =j$. For example, the first three Bell polynomials are given by \begin{equation}\label{eq:Bell_poly_first} B_1(z_1)=z_1, \quad B_2(z_1,z_2) = z_1^2 + z_2, \quad B_3(z_1,z_2,z_3) = z_1^3+3z_1z_2+z_3. \end{equation} Further, we will use the ``probabilist'' \textit{Hermite polynomials} $\Herm_n(x)$ defined by \begin{equation}\label{eq:Herm} \Herm_n(x)= \eee^{\frac 12 x^2} \left(-\frac{\dd}{\dd x}\right)^n \eee^{-\frac 12 x^2},\quad n\in\N_0. \end{equation} The first few Hermite polynomials needed for the first three terms of the expansion are \begin{align*} &\Herm_0(x)=1,\quad \Herm_1(x)= x,\quad \Herm_2(x)= x^2-1, \quad \Herm_3(x) = x^3-3x,\\ &\Herm_4(x)= x^4-6x^2+3, \quad \Herm_6(x)= x^6 - 15 x^4 + 45 x^2 - 15. \end{align*} \begin{theorem}\label{theo:stirling_edgeworth} Fix $r\in \N_0$ and a compact subset $L\subset (0,\infty)$. Uniformly over $\theta\in L$ we have $$ \lim_{n\to\infty}(\log n)^{\frac{r+1}{2}}\sup_{k=1,\ldots,n} \left|\P(K_n(\theta)=k) - \frac{\eee^{-\frac 12 x_n^2(k,\theta)}}{\sqrt{2\pi \theta \log n}} \sum_{j=0}^r \frac{H_j(x_n(k,\theta))}{(\theta \log n)^{j/2}} \right| = 0. $$ Here, $x_n(k,\theta) = \frac{k - \theta \log n}{\sqrt{\theta \log n}}$ and $H_j(x)$ is a polynomial of degree $3j$ given by \begin{align}\label{def_G_ewens} H_j(x) := H_j(x,\theta) = \frac {(-1)^j} {j!} \, \eee^{\frac 12 x^2} B_j(\widetilde {D_1},\ldots, \widetilde {D_j}) \eee^{-\frac 12 x^2}, \end{align} where $B_j$ is the $j$-th Bell polynomial and $\widetilde{D_1}, \widetilde{D_2},\ldots$ are differential operators given by \begin{align}\label{eq:D_def} \widetilde{D_j} := \widetilde{D_j}(\theta) = \frac{1}{(j+1)(j+2)} \left(\frac{\dd}{\dd x}\right)^{j+2} + \widetilde{\chi_j} (0) \left(\frac{\dd}{\dd x}\right)^{j} \end{align} with $\widetilde{\chi_j} (\beta) = - \left(\frac{\dd}{\dd \beta}\right)^j \log \Gamma(\theta\eee^{\beta})$ and $\Gamma$ denoting the Euler gamma function. \end{theorem} \begin{remark}\label{rem:H_i} It follows from~\eqref{eq:Bell_poly_first}, \eqref{def_G_ewens} and \eqref{eq:D_def} that the first three coefficients of the expansion are \begin{align*} H_0(x) &= 1,\label{eq:G0_stirling}\\ H_1(x) &= - \frac{\Gamma'(\theta)}{\Gamma(\theta)} \theta x + \frac{1}{6} \Herm_3(x),\\ H_2(x) &= \left(\theta^2 \frac{\Gamma'^2(\theta)}{\Gamma^2(\theta)} - \frac{\theta^2\Gamma''(\theta)+\theta\Gamma'(\theta)}{2\Gamma(\theta)} \right)\Herm_2(x) + \left(\frac{1}{24} - \frac{\Gamma'(\theta)}{6\Gamma(\theta)}\theta\right) \Herm_4(x)\\ &+ \frac{1}{72}\Herm_6(x). \end{align*} An expression for $\widetilde {\chi_j}(0)$ involving polygamma functions and Stirling numbers of the second kind will be given below in~\eqref{eq:chi_tilde_explicit}. The tilde in $\widetilde {D_j}$ and $\widetilde{\chi_j}$ is needed to keep the notation consistent with our more general work~\cite{Kabluchko+Marynych+Sulzbach:2016}. It is easy to check that $H_j(-x) = (-1)^j H_j(x)$ \cite[Remark~2.4]{Kabluchko+Marynych+Sulzbach:2016}. To compute $H_j(x)$ one can proceed as follows. First, express $\frac {1}{j!}B_j(\widetilde {D_{1}}, \ldots, \widetilde{D_j})$ as a polynomial in $D:= \frac{\dd}{\dd x}$ (and note that only even/odd powers of $D$ are present if $j$ is even/odd). Then replace each occurrence of $D^l$ by $\Herm_l(x)$; see~\eqref{eq:Herm} for justification. \end{remark} \begin{remark} It is possible to choose the value of $\theta$ as a function of $k$. One natural choice is $\theta = 1$ which provides a full version of Louchard's expansion~\cite[Theorem~2.1]{louchard_first_kind} (although he used a slightly different normalization in his analogue of $x_n(k,1)$ and his term $-355 x^3/144$ should be replaced by $-47 x^3/144$). Another possible choice is $\theta = k/\log n$ (so that $x_n(k,\theta)=0$), which gives a large-deviation-type expansion valid uniformly in the region $\eta^{-1} \log n < k < \eta \log n$, for fixed $\eta >1$ and $q\in\N_0$: $$ \frac{(k/\log n)^k}{(k/\log n)^{(n)}} \stirling{n}{k} = \frac{1}{\sqrt{2\pi k}}\sum_{s=0}^{q} \frac{H_{2s}(0, k/\log n)}{k^s} + o\left(\frac 1 {(\log n)^{q+1}} \right). $$ Observe that the terms with half-integer powers of $k$ are not present in the sum because $H_{2j+1}(0)=0$. Using the formula $$ \frac{\Gamma(n+\theta)}{n!} = n^{\theta-1} \left(1 + O\left(\frac 1n \right)\right) $$ yields the expansion \begin{equation}\label{eq:large_dev_expansion} \frac 1 {n!} \stirling{n}{k} = \frac{1}{ \Gamma (\theta)} n^{\theta -\theta \log \theta- 1} \left(\frac{1}{\sqrt{2\pi k}} \sum_{s=0}^{q} \frac{H_{2s}(0, \theta)}{k^s} + o\left(\frac 1 {(\log n)^{q+1}} \right)\right) \end{equation} valid as $n\to\infty$ uniformly over $k$ in the region $\theta = k/\log n \in (\eta^{-1},\eta)$. In this region, this expansion must be equivalent to Hwang's result~\cite[Theorem~2]{hwang_stirling}. It is not easy to rigorously verify this equivalence by a direct comparison, but we checked using Mathematica~9 that the first three non-trivial terms coincide. Note a misprint in the formula for the remainder term $Z_{\mu}(m,n)$ in ~\citet[Theorem~2]{hwang_stirling}: $(\log n)^m/ (m! n)$ should be replaced by $(\log n)/(mn)$. Expansion~\eqref{eq:large_dev_expansion} could be also deduced from the work of~\citet[Theorem 3.4]{feray_meliot_nikeghbali}. \end{remark} Taking sums over $k$ in Theorem~\ref{theo:stirling_edgeworth} and using the Euler-Maclaurin formula to approximate Riemann sums by integrals, one obtains that \begin{multline*} \P\left(\frac{K_n(\theta) - \theta \log n}{\sqrt{\theta \log n}} \leq x\right) = \Phi(x) \\ + \frac{\eee^{-x^2/2}}{\sqrt{2\pi\theta \log n}}\left(\frac 12 - \frac{x^2-1}{6} + \theta \frac{\Gamma'(\theta)}{\Gamma(\theta)}\right) + O\left(\frac 1 {\log n}\right), \end{multline*} uniformly in $x\in (\theta\log n)^{-1/2}(\Z-\theta \log n)$, where $\Phi(x)$ is the standard normal distribution function. The proof follows Gr{\"u}bel and Kabluchko ~\cite[Proposition~2.5]{gruebel_kabluchko_BRW} and is therefore omitted. {\citet{Yamato:2013} recently stated a slightly incorrect version of this expansion missing the term $1/2$ which comes from the Euler-Maclaurin formula.} Similarly, one can obtain further terms in the expansion of the distribution function of $(K_n(\theta)-\theta \log n) / \sqrt{\theta \log n}$. \begin{remark} Since the set $L \subseteq (0, \infty)$ in Theorem \ref{theo:stirling_edgeworth} has to be chosen compact, our results do not yield asymptotic expansions for $\P(K_n(\theta) = k)$ in the regime $k = o(\log n)$ of the same precision as Hwang's \cite[Theorems 1 and 2]{hwang_stirling}. Also, they do not extend straightforwardly to the case $k = n - O(n^\alpha)$ for $0 < \alpha < 1$ treated by Louchard \cite[Section 3] {louchard_first_kind}. A generalization of our approach to cover these regions will be content of future work. \end{remark} \subsection{Mode and maximum of the Ewens distribution} Theorem~\ref{theo:stirling_edgeworth} allows us to deduce various results on the \emph{mode} and the \emph{maximum} of the Ewens distribution. A mode is any value $k\in\{1,\ldots,n\}$ maximizing $\P(K_n(\theta) = k)$, while the maximum $M_n(\theta)$ is defined by \begin{equation*} M_n(\theta) = \max_{1\leq k\leq n} \P(K_n(\theta)=k). \end{equation*} Let $u_n(\theta)$ denote the least mode. In this context, it is important to note that, for all $\theta > 0$, the function $k \mapsto \P(K_n(\theta)=k)$ is log-concave by a theorem attributed to Newton~\cite{hammersley,sibuya}, and \begin{multline}\label{eq:unimodal} \P(K_n(\theta) = 1) < \ldots < \P(K_n(\theta) = u_n(\theta)) \\ \geq \P(K_n(\theta) = u_n(\theta)+1) > \ldots > \P(K_n(\theta) = n). \end{multline} In particular, there are at most two modes. For $\theta=1$, Erd\H{o}s~\cite{erdoes_on_hammersley}, proving a conjecture of Hammersley~\cite{hammersley}, showed that the mode is unique for all $n\geq 3$. {By~\eqref{eq:unimodal}}, uniqueness also holds for irrational $\theta$; however, for rational $\theta$, the mode need not be unique since, for example, $$ \frac 23 \stirling{3}{1} = \left(\frac 23\right)^2 \stirling{3}{2} > \left(\frac 23\right)^3 \stirling{3}{3}. $$ \begin{theorem}\label{theo:stirling_mode_maximum} Fix $\theta>0$. There exists $N_1\in\N$ such that for $n\geq N_1$, $u_n(\theta)$ is the unique mode of the Ewens distribution with parameter $\theta$. The mode $u_n(\theta)$ equals one of the numbers $\lfloor u_n^*(\theta)\rfloor$ or $\lceil u_n^*(\theta)\rceil$, where \begin{equation*} u_n^*(\theta) = \theta\log n - \frac{\theta \Gamma'(\theta)}{\Gamma(\theta)} - \frac 12 \end{equation*} and $\lfloor \cdot \rfloor$, $\lceil \cdot \rceil$ denote the floor and the ceiling functions, respectively. Write $\delta_n(\theta) := \min_{k\in\Z} |u_n^{*}(\theta) - k|$. For the maximum $M_n(\theta)$, we have $$ \sqrt{2\pi \theta \log n} \; M_n(\theta) = 1 + \frac{\theta (\log \Gamma)'(\theta) + \theta^2 (\log \Gamma)''(\theta) +1/12 - \delta_n^{2}(\theta) }{2\theta\log n} + o\left(\frac 1 {\log n}\right). $$ \end{theorem} In the case $\theta=1$, \citet{hammersley} and Erd\H{o}s \cite{erdoes_on_hammersley} derived related results for the mode. \citet{cramer} discusses statistical applications and~\citet{mezoe} provides an overview and generalizations. Theorem~\ref{theo:stirling_mode_maximum} states that the mode is one of the numbers $\lfloor \log n + \gamma -\frac 12\rfloor$ or $\lceil \log n + \gamma -\frac 12\rceil$, for sufficiently large $n$. In fact, this holds for all $n\in\N$. \begin{proposition}\label{prop:mode_stirling} $ u_n(1) \in \{\lfloor \log n + \gamma -\frac 12\rfloor, \lceil \log n + \gamma -\frac 12 \rceil \} $ for all $n\in \N$. \end{proposition} The proof uses the following formula of~\citet{hammersley}: \begin{equation}\label{eq:hammersley_mode_1} u_n(1) = \left\lfloor \log n + \gamma + \frac{\zeta(2)-\zeta(3)}{\log n + \gamma - \frac 32} + \frac{h(n)}{(\log n + \gamma - \frac 32)^2}\right\rfloor, \end{equation} for some $-1.098011 < h(n) < 1.430089$. Hwang ~\cite[Section~5.7.9]{hwang_diss} gives a more precise expansion. Erd\H{o}s \cite{erdoes_on_hammersley} observed that, for $n>189$, Hammersley's formula implies that the mode is one of the numbers $\lfloor \log (n-1) + \frac 12\rfloor $ or $\lfloor \log (n-1)+1\rfloor $. Note that his $\Sigma_{n,s}$ equals $\stirling{n+1}{n+1-s}$ and his $n-f(n)$ is $u_{n+1}(1)-1$ in our notation. The next theorem provides more precise information about the behavior of the mode. Recall that a set $A \subset \N$ has \emph{asymptotic density} $\alpha \in [0,1]$ if \begin{equation*} \lim_{n\to\infty} \frac {\# (A \cap \{1, \ldots, n\})}{n} = \alpha. \end{equation*} For $x\in\R$, let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of $x$. Let $\nint(x)$ be the integer closest to $x$ (if $\{x\}=1/2$, we agree to take $\nint (x) = \lceil x \rceil$). That is, $$ \nint(x):=\argmin_{k\in\Z}|x-k| = \left\lfloor x+\frac 12\right\rfloor. $$ \begin{theorem}\label{theo:stirling_mode_further} Fix $\theta>0$. The mode $u_n(\theta)$ of the Ewens distribution with parameter $\theta$ has the following properties: \begin{enumerate} \item [\emph{(i)}] there exists a constant $C_0>0$ such that, for all $n\in\N$ satisfying $$ \left|\{u_n^*(\theta)\} - \frac 12\right| > \frac {C_0} {\log n}, $$ the mode $u_n(\theta)$ equals $$ \nint(u_n^{\ast}(\theta)) = \left \lfloor \theta\log n - \frac{\theta \Gamma'(\theta)}{\Gamma(\theta)}\right\rfloor; $$ \item [\emph{(ii)}] there are arbitrarily long intervals of consecutive $n$'s for which $u_n(\theta)=\lceil u_n^{\ast}(\theta)\rceil$; similarly, there are arbitrarily long intervals of consecutive $n$'s for which $u_n(\theta)=\lfloor u_n^{\ast}(\theta)\rfloor$; \item [\emph{(iii)}] the set of $n\in\N$ such that $u_n(\theta)=\nint(u_n^{\ast}(\theta))$ has asymptotic density one; \item [\emph{(iv)}] there are infinitely many $n\in\N$ such that $u_n(\theta)\neq \nint(u_n^{\ast}(\theta))$. \end{enumerate} \end{theorem} The proof of part (iv) uses five terms in the Edgeworth expansion, where the first two terms influence the form of $u_n^*(\theta)$, while the remaining terms are needed for technical reasons. The idea is that the formula $u_n(\theta) = \nint(u_n^{\ast}(\theta))$ becomes wrong if the fractional part of $u_n^{*}(\theta)$ is slightly below $\frac 12$, so that higher order terms in the Edgeworth expansion decide which of the values $\lfloor u_n^*(\theta)\rfloor$ and $\lceil u_n^*(\theta)\rceil$ is the mode. Using even more terms in the expansion, it is possible to replace $u_n^*(\theta)$ by some more complicated expressions involving higher-order corrections in inverse powers of $\theta \log n$ ~\cite[Section~5.7.9]{hwang_diss}. However, it seems that there is no formula of the form $$ u_n(1) = \nint \left(\log n + a_0 + \frac {a_1}{\log n} + \cdots + \frac{a_r}{(\log n)^r}\right) $$ which is valid for all sufficiently large $n$. Finally, we would like to mention that one can easily obtain counterparts of the above results for the $B$- and $D$-analogues of Stirling numbers of the first kind. These are defined as the coefficients of $(x+1)(x+3)\cdots (x+2n-1)$ and $((x+1)(x+3)\cdots (x+2n-3)) (x+n-1)$, respectively. They appear, for example, in the study of intrinsic volumes of Weyl chambers~\cite{kabl_vys_zaporozhets_weyl_chambers}. \section{Proofs} \begin{proof}[Proof of Theorem \ref{theo:stirling_edgeworth}] The proof follows from the general Edgeworth expansion for random or deterministic profiles \cite[Theorem 2.1] {Kabluchko+Marynych+Sulzbach:2016}. We consider the sequence of ``profiles'' \begin{align*} \bL_n(k):=\P(K_n(\theta)=k) = \frac{\theta^k}{\theta^{(n)}}\stirling{n}{k} \ind_{\{k \in \{1, \ldots, n\}\}}, \end{align*} and define $$ w_n:=\theta\log n,\quad \varphi(\beta):=\eee^{\beta}-1, \quad (\beta_{-},\beta_{+})=\R, \quad \fD=\{z\in\C\colon |\text{Im } z| <\pi \}. $$ In order to apply~\cite[Theorem 2.1]{Kabluchko+Marynych+Sulzbach:2016}, we need to check Conditions A1--A4 given in the beginning of Section 2 of the cited paper. Note that \begin{align*} W_n(\beta)&:=\eee^{-\varphi(\beta)w_n}\sum_{k\in\Z}\eee^{\beta k}\bL_n(k)=n^{-\theta(\eee^{\beta}-1)}\sum_{k=1}^{n}\eee^{\beta k}\frac{\theta^k}{\theta^{(n)}}\stirling{n}{k}\\ &= n^{-\theta(\eee^{\beta}-1)}\frac{(\theta\eee^{\beta})^{(n)}}{\theta^{(n)}} = n^{-\theta (\eee^{\beta}-1)} \frac{\Gamma(\theta\eee^\beta + n) \Gamma(\theta)}{\Gamma(\theta\eee^\beta)\Gamma(\theta+n)} \ton \frac{\Gamma(\theta)}{\Gamma(\theta\eee^\beta)}=:W_{\infty}(\beta) \end{align*} locally uniformly in $\beta\in\fD$ with a rate of convergence which is polynomial in $n^{-1}$. Hence Conditions A1--A3 are satisfied. In order to check A4 it is enough to show that for every $a>0$, $r\in\N$ and every compact subset $K_1$ of $\R$ \begin{equation*} \sup_{\beta \in K_1} \sup_{a \leq u \leq \pi}\left(n^{-\theta (\eee^\beta -1)} \left|\frac{\Gamma(\theta\eee^{\beta + iu} + n)\Gamma(\theta)}{\Gamma(\theta+n)\Gamma(\theta\eee^{\beta+iu})}\right|\right)=o(\log^{-r} n),\quad n\to\infty. \end{equation*} But this easily follows from \begin{align*} \lefteqn{\sup_{\beta \in K_1} \left(n^{-\theta (\eee^\beta -1)} \sup_{a \leq u \leq \pi} \left|\frac{\Gamma(\theta\eee^{\beta + iu} + n)\Gamma(\theta)}{\Gamma(\theta+n)\Gamma(\theta\eee^{\beta+iu})}\right|\right)}\\ &\leq C\sup_{\beta \in K_1} \left(n^{-\theta (\eee^\beta -1)} \sup_{a \leq u \leq \pi} \left|\frac{\Gamma(\theta\eee^{\beta + iu} + n)}{\Gamma(\theta+n)}\right|\right)\leq C_1 \sup_{\beta \in K_1} n^{\theta \eee^{\beta}(\cos a-1)}, \end{align*} with constants $C,C_1$ depending on $K_1$, $\theta$ and $a$. Therefore, Theorem 2.1 of \cite{Kabluchko+Marynych+Sulzbach:2016} is applicable for the Ewens distribution with arbitrary fixed $\theta>0$. In particular, for $\theta=1$, we obtain \begin{align}\label{proof_thm_main4} (\log n)^{\frac{r+1}{2}} \sup_{\beta\in K} \sup_{1 \leq k \leq n} \left| \frac{\Gamma(\eee^{\beta}) \eee^{\beta k}}{n^{\eee^{\beta}-1} n!}\stirling{n}{k} - \frac{\eee^{-\frac 12 x_n^2(k,\eee^{\beta})}}{\sqrt{2 \pi \eee^{\beta} \log n}} \sum_{j=0}^r \frac{G_j(x_n(k,\eee^{\beta}); \beta)}{(\log n)^{j/2}} \right| \ton 0, \end{align} where $K$ is a compact subset of $\R$, and the polynomials $G_0, G_1, \ldots$ are defined as in Theorem 2.1 of \cite{Kabluchko+Marynych+Sulzbach:2016}: for $j \in \N_0$, we have \begin{equation}\label{eq:def_G} G_j(x; \beta) = \frac{(-1)^j}{j!} \eee^{\frac 1 2 x^2} B_j(D_1, \ldots, D_j) \eee^{-\frac 1 2 x^2} \end{equation} with the differential operators \begin{equation}\label{eq:D_def_no_tilde} D_j := D_j(\beta) = \eee^{-\frac 1 2 \beta j}\left( \frac{1}{(j+1)(j+2)} \left ( \frac{\dd}{\dd x}\right)^{j+2} + \chi_j(\beta) \left(\frac{\dd}{\dd x}\right)^{j}\right), \end{equation} where $$\chi_j(\beta) = - \left(\frac{\dd}{\dd \beta}\right)^{j} \log \Gamma (\eee^\beta).$$ Now, if $L \subseteq (0,\infty)$ is compact, then $K:=\log L$ is compact in $\R$. Applying \eqref{proof_thm_main4} with $K= \log L$ and $\beta=\log \theta\in K$, we obtain \begin{align*} (\log n)^{\frac{r+1}{2}} \sup_{\theta\in L} \sup_{1 \leq k \leq n} \left| \frac{\Gamma(\theta) \theta^k}{n^{\theta-1} n!}\stirling{n}{k} - \frac{\eee^{-\frac 12 x_n^2(k,\theta)}}{\sqrt{2 \pi \theta \log n}} \sum_{j=0}^r \frac{G_j(x_n(k,\theta); \log \theta)}{(\log n)^{j/2}} \right| \ton 0. \end{align*} By Stirling's formula, uniformly in $\theta \in L$, $n \in\N$ and $1 \leq k \leq n$, we have \begin{align*} \frac{\Gamma(\theta) \theta^k}{n^{\theta-1} n!}\stirling{n}{k} = \frac{\theta^k}{\theta^{(n)}} \stirling{n}{k} (1 + O(n^{-1}))=\frac{\theta^k}{\theta^{(n)}} \stirling{n}{k}+ O(n^{-1}). \end{align*} We conclude the proof by noting that $G_j(x; \log \theta)=\theta^{-j/2}H_j(x)$ which follows directly from $\widetilde {\chi_j}(0) = \chi_j(\log \theta)$. Indeed, by comparing~\eqref{eq:D_def} and~\eqref{eq:D_def_no_tilde}, we obtain $$ D_j(\log \theta) = \theta ^{-j/2} \widetilde {D_j}(\theta), $$ which implies that $$ B_j(D_1(\log \theta),\ldots,D_j(\log \theta)) = \theta^{-j/2} B_j(\widetilde {D_1}(\theta), \ldots, \widetilde {D_j}(\theta)) $$ since $B_j(z_1,\ldots,z_j)$ is a sum of terms of the form $c \cdot z_1^{i_1}z_2^{i_2} \cdots z_j^{i_j}$ with $1i_1+2i_2+ \cdots + ji_j =j$; see~\eqref{eq:bell_poly_def}. Comparing~\eqref{def_G_ewens} and~\eqref{eq:def_G}, we obtain the required identity $G_j(x; \log \theta)=\theta^{-j/2}H_j(x)$. To see that $\widetilde {\chi_j}(0) = \chi_j(\log \theta)$, one can easily show by induction over $j \geq 1$ that, both \begin{equation*} \chi_j(\beta) = - \sum_{\ell = 1}^{j} \stirlingb{j}{\ell} \psi^{(\ell -1)}(\eee^\beta) \eee^{ \ell \beta}, \end{equation*} and \begin{equation}\label{eq:chi_tilde_explicit} \widetilde{\chi_j}(\beta) = - \sum_{\ell = 1}^{j} \stirlingb{j}{\ell} \psi^{(\ell-1)}(\theta \eee^\beta) (\theta\eee^{\beta})^{\ell}. \end{equation} Here $\psi^{(j)}(x) = (\log \Gamma(x))^{(j+1)}$ denotes the polygamma function and $\stirlingb{n}{k}$ is the Stirling number of the second kind satisfying the recurrence $$ \stirlingb{n+1}{k} = \stirlingb{n}{k-1} + k \stirlingb{n}{k}, \quad 1 \leq k \leq n, \;\; n \in\N, $$ with initial conditions $\stirlingb{0}{0} = 1$, $\stirlingb{n}{0} = \stirlingb{0}{n} = 0$. \end{proof} \begin{proof}[Proof of Theorem \ref{theo:stirling_mode_maximum}] It follows from Theorems~2.10 in \cite{Kabluchko+Marynych+Sulzbach:2016} that for sufficiently large $n$, the maximizers of the function $k\mapsto \P(K_n(\theta) = k)$ must be of the form $\lfloor u_n^* \rfloor$ or $\lceil u_n^*\rceil$. Next we prove that the maximizer is unique (for sufficiently large $n$) by following a method of Erd\H{o}s \cite{erdoes_on_hammersley} who considered the case $\theta=1$. {Thanks to~\eqref{eq:unimodal},} the uniqueness is evident if $\theta$ is irrational. Hence, we assume that $\theta = Q_1/Q_2$ is rational with $Q_1,Q_2$ being integer. We have, by~\eqref{eq:stirling_def}, $$ \stirling{n}{k} = \sum_{1\leq a_1 <\ldots < a_{n-k} \leq n-1} a_1\cdots a_{n-k}. $$ Put $k_n = \lceil u_n^*(\theta)\rceil = \theta \log n + O(1)$ as $n\to\infty$. {By~\eqref{eq:unimodal}, it is sufficient} to show that \begin{equation}\label{eq:mode_unique_have_to_show} \theta^{k_n} \stirling {n}{k_n} \neq \theta^{k_n-1} \stirling {n}{k_n-1}. \end{equation} By Erd{\H{o}}s' argument relying on the prime number theorem with an appropriate error term \cite[p.~233]{erdoes_on_hammersley}, for all sufficiently large $n$, there is a prime number $p$ satisfying {$(n-1)/k_n < p < (n-1)/(k_n-1)$.} Then, $$ \stirling {n}{k_n} \not \equiv 0 \pmod p, \quad \stirling {n}{k_n - 1} \equiv 0 \pmod p $$ because in the representation of the former Stirling number all products except one are divisible by $p$, whereas in the latter all products are divisible by $p$. If $n$ is large, $p$ is not among the prime factors of $Q_1$ and $Q_2$. Hence~\eqref{eq:mode_unique_have_to_show} follows and the mode of $K_n(\theta)$ is unique. Finally, the formula for $M_n$ follows from Theorem~2.13 of~\cite{Kabluchko+Marynych+Sulzbach:2016}. \end{proof} \begin{proof}[Proof of Proposition~\ref{prop:mode_stirling}] Recall Hammersley's formula \eqref{eq:hammersley_mode_1}: \begin{equation*} u_n(1) = \left\lfloor \log n + \gamma + \frac{\zeta(2)-\zeta(3)}{\log n + \gamma - \frac 32} + \frac{h(n)}{(\log n + \gamma - \frac 32)^2}\right\rfloor \end{equation*} with some $-1.1 < h(n) < 1.44$. It is easy to check that $$ \frac{\zeta(2)-\zeta(3)}{x} - \frac {1.1}{x^2} > -\frac 12 \text{ and } \frac{\zeta(2)-\zeta(3)}{x} + \frac {1.44}{x^2} <\frac 12 $$ for $x>2.5$. Hence, the proposition is true for $\log n + \gamma - \frac 32 > 2.5$, that is for $n\geq 31$. For $n=1,2,\ldots,30$ the statement is easy to verify using Mathematica~9. \end{proof} \begin{proof}[Proof of Theorem \ref{theo:stirling_mode_further} (i) and (ii)] Part (i) follows essentially from Theorem~2.10 in~\cite{Kabluchko+Marynych+Sulzbach:2016} and its proof. Namely, by \cite[Equation~(90)]{Kabluchko+Marynych+Sulzbach:2016}, for $k= k(n)= u_n^*(\theta) + g \in \Z$ with $g=O(1)$, we have $$ \sqrt{2\pi \theta \log n} \left(\P(K_n(\theta) = k+1) - \P(K_n(\theta) = k)\right) = -\frac {2g+1}{2\theta \log n} + o\left(\frac 1 {\log n} \right). $$ The same relation, but with a better remainder term $O(\frac 1 {\log^2 n})$, follows from~\eqref{proof_iv1} which we shall prove below. Taking $g=-\{u_n^*(\theta)\}$, so that $k= \lfloor u_n^*(\theta)\rfloor$ and $k+1=\lceil u_n^*(\theta)\rceil$, yields \begin{multline*} \P(K_n(\theta) = \lceil u_n^*(\theta)\rceil) - \P(K_n(\theta) = \lfloor u_n^*(\theta)\rfloor) \\= \frac 1 {\sqrt{2\pi \theta \log n}} \left(\frac {\{u_n^*(\theta)\}-\frac 12}{\theta \log n} + O\left(\frac 1 {\log^2 n} \right)\right). \end{multline*} It follows that there is a sufficiently large constant $C_0>0$ such that, if $\{u_n^*(\theta)\} > \frac 12 + \frac {C_0}{\log n}$, then the right-hand side is positive, and the mode equals $\lceil u_n^*(\theta)\rceil$. Similarly, if $\{u_n^*(\theta)\} < \frac 12 - \frac {C_0}{\log n}$, then the right-hand side is negative, and the mode equals $\lfloor u_n^*(\theta)\rfloor$. The proof of part (ii) follows immediately from part (i) and the fact that, for every fixed $L>0$, we have $\log (n+L)-\log n\to 0$ as $n\to\infty$. \end{proof} \begin{proof}[Proof of Theorem \ref{theo:stirling_mode_further} (iii)] In view of part (i) it suffices to show that $$\limsup_{\eps \to 0} \limsup_{n \to \infty} \frac{\#\{1 \leq k \leq n: \text{dist}(u_k^*(\theta), \Z + 1/2) < \eps \} }{n} = 0,$$ which, in turn, follows from the fact that \begin{equation}\label{proof_iii} \limsup_{\eps \to 0} \limsup_{n \to \infty} \frac{\#\{1 \leq k \leq n: \text{dist}(\log k, \alpha \Z + \beta) < \eps \} }{n} = 0, \end{equation} for all $\alpha>0$ and $\beta \in \R$. Equation~\eqref{proof_iii} would be true if the sequence of fractional parts of $\alpha^{-1}\log k$, $k\in\N$, were uniformly distributed on $[0,1]$. However, the latter claim is unfortunately not true \cite[Examples~2.4 and~2.5, pp.~8--9]{kuipers_niederreiter_book}. Let us prove \eqref{proof_iii}. We have, assuming that $\varepsilon<\alpha/2$, \begin{align*} \#\{1 \leq k \leq n: \text{dist}(\log k, & \alpha \Z + \beta) < \eps \} =\sum_{k=1}^{n} \# \{ j \in \Z : \text{dist}(\log k, \alpha j + \beta) < \eps\}\\ & = \sum_{j\in\Z}\#\{1 \leq k \leq n: \eee^{\alpha j+\beta-\eps} < k < \eee^{\alpha j+\beta+\eps} \} \\ & \leq \sum_{j\in\Z} \# \{k \in \N : \eee^{\alpha j+\beta-\eps}\vee 1 \leq k \leq \eee^{\alpha j+\beta+\eps}\wedge n \}. \end{align*} The summand on the right-hand side is the number of integers in the interval $[\eee^{\alpha j+\beta-\eps}\vee 1,\,\eee^{\alpha j+\beta+\eps}\wedge n]$ (which is empty if either $\eee^{\alpha j+\beta-\eps}>n$ or $\eee^{\alpha j+\beta+\eps}<1$). Hence, it is bounded from above by $\left(\eee^{\alpha j+\beta+\eps}\wedge n - \eee^{\alpha j+\beta-\eps}\vee 1+1\right)_{+}$. Therefore, $$ \#\{1 \leq k \leq n: \text{dist}(\log k, \alpha \Z + \beta) < \eps \} \leq \sum_{j\in\Z}\left(\eee^{\alpha j+\beta+\eps}\wedge n - \eee^{\alpha j+\beta-\eps}\vee 1+1\right)_{+}. $$ Further, \begin{align*} &\sum_{j\in\Z}\left(\eee^{\alpha j+\beta+\eps}\wedge n - \eee^{\alpha j+\beta-\eps}\vee 1+1\right)_{+}\\ &=\sum_{j\in\Z}\eee^{\alpha j+\beta+\eps}\ind_{\{\alpha j+\beta+\varepsilon<0\}}+\sum_{j\in\Z}\eee^{\alpha j+\beta+\eps}\ind_{\{\alpha j+\beta-\varepsilon<0,0\leq \alpha j+\beta+\varepsilon<\log n\}}\\ &+\sum_{j\in\Z}n\ind_{\{\alpha j+\beta-\varepsilon<0,\log n\leq \alpha j+\beta+\varepsilon\}}\\ &+\sum_{j\in\Z}\left(\eee^{\alpha j+\beta+\eps} - \eee^{\alpha j+\beta-\eps}+1\right)\ind_{\{\alpha j+\beta-\varepsilon\geq 0,\alpha j+\beta+\varepsilon<\log n\}}\\ &+\sum_{j\in\Z}\left(n - \eee^{\alpha j+\beta-\eps}+1\right)_{+}\ind_{\{\alpha j+\beta-\varepsilon\geq 0,\log n\leq \alpha j+\beta+\varepsilon\}}. \end{align*} Note that the first series converges, the second contains at most one summand since we assume $\varepsilon<\alpha/2$, and the third vanishes for $n$ large enough. It can be checked that \begin{align*} \sum_{j\in\Z}\left(\eee^{\alpha j+\beta+\eps} - \eee^{\alpha j+\beta-\eps}+1\right)\ind_{\{\alpha j+\beta-\varepsilon\geq 0,\alpha j+\beta+\varepsilon<\log n\}}\leq C(\alpha,\beta)(\eee^{\beta+\eps}-\eee^{\beta-\eps})n \end{align*} with an absolute constant $C(\alpha,\beta)$. Further, for $n$ sufficiently large, we have \begin{align*} \sum_{j\in\Z}\left(n - \eee^{\alpha j+\beta-\eps}+1\right)_{+}\ind_{\{\alpha j+\beta-\varepsilon\geq 0,\log n\leq \alpha j+\beta+\varepsilon\}}\leq n(1-\eee^{-2\eps})+1. \end{align*} Putting pieces together gives \eqref{proof_iii}. \end{proof} \begin{proof}[Proof of Theorem \ref{theo:stirling_mode_further} (iv)] Recall the notation $w_n=\theta\log n$ and $x_n(k) = x_n(k,\theta)=(k-w_n)/\sqrt{w_n}$. Using Theorem~\ref{theo:stirling_edgeworth} with $r=4$, we obtain \begin{multline*} \sqrt{2\pi w_n}\, \P(K_n(\theta)=k)=\eee^{-\frac 12 {x_n^2(k)}}\\ \times\left(1+\frac{H_1(x_n(k))}{w_n^{1/2}}+\frac{H_2(x_n(k))}{w_n}+\frac{H_3(x_n(k))}{w_n^{3/2}}+\frac{H_4(x_n(k))}{w_n^2}+o\left(\frac{1}{\log^2 n}\right)\right), \end{multline*} {as $n \to\infty$ uniformly in $1 \leq k \leq n$.} Now let $k=\theta\log n + a$, where $a=O(1)$ as $n\to\infty$, so that $x_n(k)=a/w_n^{1/2}$. We have \begin{align*} H_1(x_n(k))&=A_{11}(\theta)\frac{a}{w_n^{1/2}}+A_{12}(\theta)\frac{a^3}{w_n^{3/2}},\\ H_2(x_n(k))&=A_{21}(\theta)+A_{22}(\theta)\frac{a^2}{w_n}+o\left(\frac{1}{w_n}\right),\\ H_3(x_n(k))&=A_{31}(\theta)\frac{a}{w_n^{1/2}}+o\left(\frac{1}{w_n^{1/2}}\right),\\ H_4(x_n(k))&=A_{41}(\theta)+o(1), \end{align*} where $A_{11}(\theta),\ldots,A_{41}(\theta)$ are some polynomials in $\widetilde {\chi_1}(0), \widetilde {\chi_2}(0),\widetilde {\chi_3}(0)$ and $\widetilde{\chi_4}(0)$; see Remark~\ref{rem:H_i}. Plugging these expressions into the asymptotic expansion above and using the {expansion} $\eee^y=1+y+y^2/2+o(y^2)$, as $y\to 0$, yields $$ \sqrt{2\pi w_n}\,\P(K_n(\theta)=k)=1-\left(\frac{a^2}{2}-A_{11}(\theta)a-A_{21}(\theta)\right)\frac{1}{w_n}+\frac{P_{\theta}(a)}{w_n^2}+o\left(\frac{1}{\log^2 n}\right), $$ where $$ P_{\theta}(a):=\frac{1}{8}a^4+\left(A_{12}(\theta)-\frac{1}{2}A_{11}(\theta)\right)a^3+\left(A_{22}(\theta)-\frac{1}{2}A_{21}(\theta)\right)a^2+A_{31}(\theta)a + A_{41}(\theta). $$ Now let us write $k=\theta\log n +a^{\ast} + g$, where $a^{\ast}:=A_{11}(\theta)=-\frac{\theta\Gamma'(\theta)}{\Gamma(\theta)}-\frac 12$, yielding \begin{multline}\label{proof_iv1} \sqrt{2\pi w_n}\,\P(K_n(\theta)=k) = 1-\left(\frac{g^2-(a^{\ast})^2}{2}-A_{21}(\theta)\right)\frac{1}{w_n}\\ +\frac{P_{\theta}(a^{\ast}+g)}{w_n^2}+o\left(\frac{1}{\log^2 n}\right). \end{multline} We are interested in $g$ being either $\lfloor u_n^{\ast}(\theta)\rfloor -u_n^{\ast}(\theta)=:g_n^{\prime}$ or $\lceil u_n^{\ast}(\theta)\rceil -u_n^{\ast}(\theta)=:g_n^{\prime\prime}$. {Let $M$ be the set of natural numbers $n$ with $\{u_n^{\ast}(\theta)\}<1/2<\{u_{n+1}^{\ast}(\theta)\}$.} Note that $M$ has infinitely many elements because $\log n\to \infty$ and $\log (n+1)-\log n\to 0$. {In the remainder of the proof, we always consider $n \in M$.} Since $u^{\ast}_{n+1}(\theta)-u^{\ast}_n(\theta)=O(n^{-1})$, we have \begin{align*} g_n^{\prime}=-1/2+O(n^{-1}), \quad g_n^{\prime\prime}=1/2+O(n^{-1}). \end{align*} Putting $k = \lfloor u_n^{\ast}(\theta)\rfloor$ into \eqref{proof_iv1} yields \begin{align*} &\sqrt{2\pi w_n}\,\P(K_n(\theta)=\lfloor u_n^{\ast}(\theta)\rfloor)\\ &=1-\left(\frac{(g_n^{\prime})^2-(a^{\ast})^2}{2}-A_{21}(\theta)\right)\frac{1}{w_n}+\frac{P_{\theta}(a^{\ast}+g_n^{\prime})}{w_n^2}+o\left(\frac{1}{\log^2 n}\right)\\ &=1-\left(\frac{1-4(a^{\ast})^2}{8}-A_{21}(\theta)\right)\frac{1}{w_n}+\frac{P_{\theta}(a^{\ast}-1/2)}{w_n^2}+o\left(\frac{1}{\log^2 n}\right). \end{align*} Analogously, putting $k = \lceil u_n^{\ast}(\theta)\rceil$ gives \begin{align*} &\sqrt{2\pi w_n}\,\P(K_n(\theta)=\lceil u_n^{\ast}(\theta)\rceil)\\ &=1-\left(\frac{1-4(a^{\ast})^2}{8}-A_{21}(\theta)\right)\frac{1}{w_n}+\frac{P_{\theta}(a^{\ast}+1/2)}{w_n^2}+o\left(\frac{1}{\log^2 n}\right). \end{align*} For sufficiently large $n$ the mode $u_n(\theta)$ equals either $\lfloor u_n^{\ast}(\theta)\rfloor$ or $\lceil u_n^{\ast}(\theta)\rceil$ depending on the sign of $$ s^{\ast}(\theta):=P_{\theta}(a^{\ast}+1/2)-P_{\theta}(a^{\ast}-1/2). $$ In the following we shall show that $s^{\ast}(\theta)>0$, hence $u_n(\theta)=\lceil u_n^{\ast}(\theta)\rceil$, while $\nint (u_{n}(\theta)) = \lfloor u_n^{\ast}(\theta)\rfloor$, so that $u_{n}(\theta)\neq \nint(u_n^{\ast}(\theta))$. Recalling the polygamma function $\psi^{(m)} (\theta) = (\log \Gamma(\theta))^{(m+1)}$, the authors \cite{mathematica} checked with the help of Mathematica~9 that $$ s^{\ast}(\theta)=\frac{\theta^2}{2}\left(2 \psi^{(1)}(\theta) +\theta\psi^{(2)}(\theta) \right). $$ Using the well-known formula for the polygamma function \cite[6.4.10]{Abramowitz+Stegun:1964} $$ \psi^{(m)} (\theta) = (\log \Gamma(\theta))^{(m+1)}=(-1)^{m+1}m!\sum_{k=0}^{\infty}\frac{1}{(\theta+k)^{m+1}},\quad -\theta\notin \N_0,\quad m \geq 1, $$ we finally obtain $$ s^{\ast}(\theta)=\theta^2\sum_{k=1}^{\infty}\frac{k}{(\theta+k)^3},\quad \theta>0, $$ yielding positivity of $s^{\ast}(\theta)$ for all $\theta>0$. The proof of part (iv), as well as of the whole theorem, is complete. \end{proof} \begin{remark} For $\theta=1$ we have $s^*(1) = \zeta(2) - \zeta(3)$, a term appearing in Hammersley's formula~\eqref{eq:hammersley_mode_1}. In fact, in the special case $\theta=1$ part (iv) could be deduced directly from~\eqref{eq:hammersley_mode_1}. \end{remark} \section{Acknowledgements} The work of Alexander Marynych was supported by a Humboldt Research Fellowship of the Alexander von Humboldt Foundation. The work of Henning Sulzbach was supported by a Feodor Lynen Research Fellowship of the Alexander von Humboldt Foundation. \bibliographystyle{plainnat} \begin{thebibliography}{25} \providecommand{\natexlab}[1]{#1} \providecommand{\url}[1]{\texttt{#1}} \expandafter\ifx\csname urlstyle\endcsname\relax \providecommand{\doi}[1]{doi: #1}\else \providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi \bibitem[Abramowitz and Stegun(1964)]{Abramowitz+Stegun:1964} M. Abramowitz and I. Stegun, \newblock \emph{Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables}, volume~55 of National Bureau of Standards Applied Mathematics Series, 1964. \bibitem[Bingham(1988)]{bingham_tauberian} N.~H. Bingham, \newblock Tauberian theorems for {J}akimovski and {K}aramata-{S}tirling methods, \newblock \emph{Mathematika} \textbf{35} (1988), 216--224. \bibitem[Cramer(2000)]{cramer} E. Cramer, \newblock Asymptotic estimators of the sample size in a record model, \newblock \emph{Statist. Papers} \textbf{41} (2000), 159--171. \bibitem[Durrett(2008)]{durrett_DNA} R. Durrett, \newblock \emph{Probability Models for {DNA} Sequence Evolution}, \newblock Probability and its Applications, Springer, second edition, 2008. \bibitem[Erd{\"o}s(1953)]{erdoes_on_hammersley} P. Erd\H{o}s, \newblock On a conjecture of {H}ammersley, \newblock \emph{J. London Math. Soc.} \textbf{28} (1953), 232--236. \bibitem[Ewens and Tavar{\'e}(1997)]{ewens_tavare} W. Ewens and S. Tavar{\'e}, \newblock Multivariate {E}wens distribution. \newblock In N.~L. Johnson, S.~Kotz, and N.~Balakrishnan, editors, \emph{Discrete Multivariate Distributions}, Wiley Series in Probability and Statistics: Applied Probability and Statistics. John Wiley \& Sons, 1997. \bibitem[Feller(1945)]{feller1945} W. Feller, \newblock The fundamental limit theorems in probability, \newblock \emph{Bull. Amer. Math. Soc.} \textbf{51} (1945), 800--832. \bibitem[F{\'e}ray et~al.(2015)F{\'e}ray, M{\'e}liot, and Nikeghbali]{feray_meliot_nikeghbali} V. F{\'e}ray, P.-L. M{\'e}liot, and A. Nikeghbali, \newblock Mod--{$\phi$} convergence, I: {N}ormality zones and precise deviations, preprint, 2015, \newblock \url{http://arxiv.org/abs/1304.2934}. \bibitem[Gr{\"u}bel and Kabluchko(2015)]{gruebel_kabluchko_BRW} R. Gr{\"u}bel and Z. Kabluchko, \newblock Edgeworth expansions for profiles of lattice branching random walks, \newblock \emph{to appear in Ann. Inst. H. Poincar{\'e}}, 2015, \url{http://arxiv.org/abs/1503.04616}. \bibitem[Hammersley(1950)]{hammersley_restricted} J. M. Hammersley, \newblock On estimating restricted parameters, \newblock \emph{J. Roy. Statist. Soc. Ser. B.} \textbf{12} (1950), 192--229, discussion, 230--240. \bibitem[Hammersley(1951)]{hammersley} J. M. Hammersley, \newblock The sums of products of the natural numbers, \newblock \emph{Proc. London Math. Soc. (3)} \textbf{1} (1951), 435--452. \bibitem[Hwang(1994)]{hwang_diss} H.-K. Hwang, \newblock Th{\'e}or{\`{e}}mes limites pour les structures combinatories et les fonctions arithm{\'e}tiques, \newblock Ph.D. thesis, 1994, \url{http://algo.stat.sinica.edu.tw/hk/?p=80}. \bibitem[Hwang(1995)]{hwang_stirling} H.-K. Hwang, \newblock Asymptotic expansions for the {S}tirling numbers of the first kind, \newblock \emph{J. Combin. Theory Ser. A} \textbf{71} (1995), 343--351. \bibitem[Kabluchko et~al.(2015)Kabluchko, Vysotsky, and Zaporozhets]{kabl_vys_zaporozhets_weyl_chambers} Z. Kabluchko, V. Vysotsky, and D. Zaporozhets, \newblock Convex hulls of random walks, hyperplane arrangements, and {W}eyl chambers, preprint, 2015, \newblock \url{http://arxiv.org/abs/1510.04073}. \bibitem[Kabluchko et~al.(2016)Kabluchko, Marynych, and Sulzbach]{mathematica} Z. Kabluchko, A. Marynych, and H. Sulzbach, \newblock Mathematica notebook, 2016, \newblock \url{http://www.math.uni-muenster.de/statistik/kabluchko/files/stirling.pdf} and \url{http://www.math.uni-muenster.de/statistik/kabluchko/files//stirling.nb}. \bibitem[Kabluchko et~al.(2016+)Kabluchko, Marynych, and Sulzbach]{Kabluchko+Marynych+Sulzbach:2016} Z. Kabluchko, A. Marynych, and H. Sulzbach, \newblock General {E}dgeworth expansions with applications to profiles of random trees, preprint, 2016, \newblock \url{http://arxiv.org/abs/1606.03920}, version with full proofs at \url{http://www.math.uni-muenster.de/statistik/kabluchko/files/edgeworth_full.pdf}. \bibitem[{Kuipers} and {Niederreiter}(1974)]{kuipers_niederreiter_book} L. {Kuipers} and H. {Niederreiter}, \newblock \emph{Uniform Distribution of Sequences}, \newblock Pure and Applied Mathematics, \newblock Wiley-Interscience, 1974. \bibitem[Louchard(2010)]{louchard_first_kind} G. Louchard, \newblock Asymptotics of the {S}tirling numbers of the first kind revisited: a saddle point approach, \newblock \emph{Discrete Math. Theor. Comput. Sci.} \textbf{12} (2010), 167--184. \bibitem[Mez{\H{o}}(2008)]{mezoe} I. Mez{\H{o}}, \newblock On the maximum of {$r$}-{S}tirling numbers, \newblock \emph{Adv. in Appl. Math.} \textbf{41} (2008), 293--306. \bibitem[Moser and Wyman(1958)]{moser_wyman} L. Moser and M. Wyman, \newblock Asymptotic development of the {S}tirling numbers of the first kind, \newblock \emph{J. London Math. Soc.} \textbf{33} (1958), 133--146. \bibitem[R{\'e}nyi(1962)]{renyi} A. R{\'e}nyi, \newblock On the outliers of a series of observations, \newblock \emph{Magyar Tud. Akad. Mat. Fiz. Oszt. K\"ozl.} \textbf{12} (1962), 105--121. \bibitem[Sibuya(1988)]{sibuya} M. Sibuya, \newblock Log-concavity of {S}tirling numbers and unimodality of {S}tirling distributions, \newblock \emph{Ann. Inst. Statist. Math.} \textbf{40} (1988), 693--714. \bibitem[Temme(1993)]{temme} N. M. Temme, \newblock Asymptotic estimates of {S}tirling numbers, \newblock \emph{Stud. Appl. Math.} \textbf{89} (1993), 233--243. \bibitem[Wilf(1993)]{wilf} H. S. Wilf, \newblock The asymptotic behavior of the {S}tirling numbers of the first kind, \newblock \emph{J. Combin. Theory Ser. A} \textbf{64} (1993), 344--349. \bibitem[Yamato(2013)]{Yamato:2013} H. Yamato, \newblock Edgeworth expansions for the number of distinct components associated with the {E}wens sampling formula, \newblock \emph{J. Japan Statist. Soc.} \textbf{43} (2013), 17--28. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2010 {\it Mathematics Subject Classification}: Primary 11B73; Secondary 60C05, 41A60, 60F05, 60F10. \noindent \emph{Keywords: } asymptotic expansion, Ewens distribution, Stirling number of the first kind, mode of a distribution. \bigskip \hrule \bigskip \noindent (Concerned with sequences \seqnum{A008275}, \seqnum{A048994}, \seqnum{A132393}.) \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received September 30 2016. Revised version received November 21 2016. Published in {\it Journal of Integer Sequences}, November 25 2016. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}