%%% sj239.tex  mahon.tex:
%%% The Mahonian Probability Distribution on Words is Asymptotically Normal
%%% by E. Rodney Canfield, Svante Janson, and Doron Zeilberger (17 pages)
%%% a LaTeX file

\documentclass[11pt,reqno,tbtags]{amsart}
%\usepackage{amsmath}
\usepackage{amssymb}
%\usepackage[latin1]{inputenc}   om svenska bokst}ver
\usepackage{upref}
%\usepackage[notcite,notref]{showkeys} % shows labels
\usepackage{natbib}
\bibpunct[, ]{[}{]}{;}{n}{,}{,}

%\addtolength{\textwidth}{10mm}

%\linespread{1.6}  % double spaced

\numberwithin{equation}{section}

%\setcounter{secnumdepth}{1}

\allowdisplaybreaks


%%%%% THEOREMS ETC %%%%%

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conj}[theorem]{Conjecture}
%\newtheorem{Alg}[theorem]{Algorithm}

\theoremstyle{definition}
\newtheorem{example}[theorem]{Example}
%\newtheorem*{definition}{Definition}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{remark}[theorem]{Remark}
%\newtheorem*{personal}{Personal remark}
\newtheorem{tom}{}
\newtheorem*{acks}{Acknowledgements}
\newtheorem*{ack}{Acknowledgement}

\theoremstyle{remark}

%\newenvironment{acks}{\section*{Acknowledgement}}{}
%\newenvironment{algo}{\begin{enumerate}}{\end{enumerate}}

\newenvironment{romenumerate}[1][0pt]{% optional argument changes indentation
\addtolength{\leftmargini}{#1}\begin{enumerate}% gives (i), (ii) etc.
 \renewcommand{\labelenumi}{\textup{(\roman{enumi})}}%
 \renewcommand{\theenumi}{\textup{(\roman{enumi})}}%
 }{\end{enumerate}}

\newcounter{oldenumi}
\newenvironment{romenumerateq}% continues numbering from previous romenumerate
{\setcounter{oldenumi}{\value{enumi}}
\begin{romenumerate} \setcounter{enumi}{\value{oldenumi}}}
{\end{romenumerate}}



% thmenumerate gives (i) run in after ``Theorem ...''
% now works with \label too, but needs '%' after \label{...}%
\newcounter{thmenumerate}
\newenvironment{thmenumerate}
{\setcounter{thmenumerate}{0}%
 \renewcommand{\thethmenumerate}{\textup{(\roman{thmenumerate})}}%
 \def\item{\par% \ifnum\thethmenumerate=0\else\par\fi %\noindent\fi
 \refstepcounter{thmenumerate}\textup{(\roman{thmenumerate})\enspace}}
}
{}

\newcounter{romxenumerate}   %less indented than standard.
\newenvironment{romxenumerate}
  {\begin{list}
    {\upshape(\roman{romxenumerate})}
    {\addtolength{\leftmargin}{-10pt} %ad lib 
     \addtolength{\labelwidth}{-10pt}
%     \addtolength{\labelsep}{0pt}
     %\setlength{\rightmargin}{0pt}
%\def\makelabel##1{K##1}
 \renewcommand{\theromxenumerate}{\textup{(\roman{romxenumerate})}}
     \usecounter{romxenumerate}} 
% \renewcommand{\theromxenumerate}{\textup{(\roman{romxenumerate})}}
}
  {\end{list}}


\newcounter{xenumerate}   %no left indentation; thus wider lines
\newenvironment{xenumerate}
  {\begin{list}
    {\upshape(\roman{xenumerate})}
    {\setlength{\leftmargin}{0pt}
     \setlength{\rightmargin}{0pt}
     \setlength{\labelwidth}{0pt}
     \setlength{\itemindent}{\labelsep}
     \setlength{\topsep}{0pt}
     \usecounter{xenumerate}} }
  {\end{list}}

\newcommand\pfitem[1]{\par(#1):}

\newcommand\step[2]{\smallskip\noindent\emph{Step #1: #2} \noindent}


\newcommand{\refT}[1]{Theorem~\ref{#1}}
\newcommand{\refC}[1]{Corollary~\ref{#1}}
\newcommand{\refL}[1]{Lemma~\ref{#1}}
\newcommand{\refR}[1]{Remark~\ref{#1}}
\newcommand{\refS}[1]{Section~\ref{#1}}
\newcommand{\refSS}[1]{Subsection~\ref{#1}}
%\newcommand{\refP}[1]{Proposition~\ref{#1}}
\newcommand{\refP}[1]{Problem~\ref{#1}}
\newcommand{\refD}[1]{Definition~\ref{#1}}
\newcommand{\refE}[1]{Example~\ref{#1}}
\newcommand{\refF}[1]{Figure~\ref{#1}}
\newcommand{\refApp}[1]{Appendix~\ref{#1}}
\newcommand{\refConj}[1]{Conjecture~\ref{#1}}
\newcommand{\refTab}[1]{Table~\ref{#1}}
\newcommand{\refand}[2]{\ref{#1} and~\ref{#2}}


% Marginal notes and labelling macros
\newcommand\marginal[1]{\marginpar{\raggedright\parindent=0pt\tiny #1}}
\newcommand\SJ{\marginal{SJ}}
%\newcommand\kolla{\marginal{CHECK! SJ}}
\newcommand\kolla{\marginal{KOLLA!}}

\newcommand\linebreakx{\unskip\marginal{$\backslash$linebreak}\linebreak}

%\def\nomarginal{%
% \def\marginal##1{\ignorespaces}%
% \overfullrule=0pt \message{*** no marginals or black boxes ***}}


\begingroup
  \count255=\time
  \divide\count255 by 60
  \count1=\count255
  \multiply\count255 by -60
  \advance\count255 by \time
  \ifnum \count255 < 10 \xdef\klockan{\the\count1.0\the\count255}
  \else\xdef\klockan{\the\count1.\the\count255}\fi
\endgroup

% further AmSTeX adaptions
\newcommand\nopf{\qed}   % for theorem without proof
\newcommand\noqed{\renewcommand{\qed}{}} % for proof with explicit \qed
\newcommand\qedtag{\tag*{\qedsymbol}}



%\DeclareMathOperator*{\sumsum}{\sum\sum}
\DeclareMathOperator*{\sumx}{\sum\nolimits^{*}}
\DeclareMathOperator*{\sumxx}{\sum\nolimits^{**}}
\DeclareMathOperator*{\sumy}{\sum\nolimits^{*}}
\DeclareMathOperator*{\sumyq}{\sum\nolimits^{*\!\!\!}}
\newcommand{\sumi}{\sum_{i=0}^\infty}
\newcommand{\sumj}{\sum_{j=0}^\infty}
\newcommand{\sumk}{\sum_{k=0}^\infty}
\newcommand{\summ}{\sum_{m=0}^\infty}
\newcommand{\sumn}{\sum_{n=0}^\infty}

\newcommand\set[1]{\ensuremath{\{#1\}}}
\newcommand\bigset[1]{\ensuremath{\bigl\{#1\bigr\}}}
\newcommand\Bigset[1]{\ensuremath{\Bigl\{#1\Bigr\}}}
\newcommand\biggset[1]{\ensuremath{\biggl\{#1\biggr\}}}
\newcommand\xpar[1]{(#1)}
\newcommand\bigpar[1]{\bigl(#1\bigr)}
\newcommand\Bigpar[1]{\Bigl(#1\Bigr)}
\newcommand\biggpar[1]{\biggl(#1\biggr)}
\newcommand\lrpar[1]{\left(#1\right)}
\newcommand\bigsqpar[1]{\bigl[#1\bigr]}
\newcommand\Bigsqpar[1]{\Bigl[#1\Bigr]}
\newcommand\biggsqpar[1]{\biggl[#1\biggr]}
\newcommand\lrsqpar[1]{\left[#1\right]}
\newcommand\xcpar[1]{\{#1\}}
\newcommand\bigcpar[1]{\bigl\{#1\bigr\}}
\newcommand\Bigcpar[1]{\Bigl\{#1\Bigr\}}
\newcommand\biggcpar[1]{\biggl\{#1\biggr\}}
\newcommand\lrcpar[1]{\left\{#1\right\}}
\newcommand\abs[1]{|#1|}
\newcommand\bigabs[1]{\bigl|#1\bigr|}
\newcommand\Bigabs[1]{\Bigl|#1\Bigr|}
\newcommand\biggabs[1]{\biggl|#1\biggr|}
\newcommand\lrabs[1]{\left|#1\right|}
\def\rompar(#1){\textup(#1\textup)}    % usage: \rompar(...)
\newcommand\xfrac[2]{#1/#2}
\newcommand\xpfrac[2]{(#1)/#2}
\newcommand\parfrac[2]{\lrpar{\frac{#1}{#2}}}
\newcommand\bigparfrac[2]{\bigpar{\frac{#1}{#2}}}
\newcommand\Bigparfrac[2]{\Bigpar{\frac{#1}{#2}}}
\newcommand\biggparfrac[2]{\biggpar{\frac{#1}{#2}}}
\newcommand\xparfrac[2]{\xpar{\xfrac{#1}{#2}}}
\newcommand\expX[1]{\exp\Bigl(#1\Bigr)}
\newcommand\expx[1]{\exp\bigl(#1\bigr)}
\newcommand\expQ[1]{e^{#1}}
\def\xexp(#1){e^{#1}}
\newcommand\ceil[1]{\lceil#1\rceil}
\newcommand\floor[1]{\lfloor#1\rfloor}
\newcommand\fract[1]{\{#1\}}
%\newcommand\setx[1]{\set{1,\dots,#1}}
\newcommand\setn{\set{1,\dots,n}}
\newcommand\ntoo{\ensuremath{{n\to\infty}}}
\newcommand\Ntoo{\ensuremath{{N\to\infty}}}
\newcommand\asntoo{\text{as }\ntoo}
\newcommand\ktoo{\ensuremath{{k\to\infty}}}
\newcommand\stoo{\ensuremath{{s\to\infty}}}
\newcommand\bmin{\wedge}
\newcommand\norm[1]{\|#1\|}
\newcommand\bignorm[1]{\bigl\|#1\bigr\|}
\newcommand\Bignorm[1]{\Bigl\|#1\Bigr\|}
\newcommand\downto{\searrow}
\newcommand\upto{\nearrow}
%\newcommand\upto{\uparrow}
\newcommand\half{\tfrac12}
\newcommand\thalf{\tfrac12}

\newcommand\iid{i.i.d.\spacefactor=1000}    
\newcommand\ie{i.e.\spacefactor=1000}
\newcommand\eg{e.g.\spacefactor=1000}
\newcommand\viz{viz.\spacefactor=1000}
\newcommand\cf{cf.\spacefactor=1000}
\newcommand{\as}{a.s.\spacefactor=1000}
\newcommand{\aex}{a.e.\spacefactor=1000}
\renewcommand{\ae}{\vu}  % som felkoll
%\newcommand\whp{\textbf{whp}}
\newcommand\whp{{w.h.p.\spacefactor=1000}}

\newcommand\ii{\mathrm{i}}

\newcommand{\tend}{\longrightarrow}
\newcommand\dto{\overset{\mathrm{d}}{\tend}}
\newcommand\pto{\overset{\mathrm{p}}{\tend}}
\newcommand\asto{\overset{\mathrm{a.s.}}{\tend}}
\newcommand\eqd{\overset{\mathrm{d}}{=}}
\newcommand\neqd{\overset{\mathrm{d}}{\neq}}
\newcommand\op{o_{\mathrm p}}
\newcommand\Op{O_{\mathrm p}}

\newcommand\bbR{\mathbb R}
\newcommand\bbC{\mathbb C}
\newcommand\bbN{\mathbb N}  %{1,2,...}
\newcommand\bbT{\mathbb T}
\newcommand\bbZ{\mathbb Z}

\newcounter{CC}
\newcommand{\CC}{\stepcounter{CC}\CCx} %new constant C_i
\newcommand{\CCx}{C_{\arabic{CC}}}     %repeats the last C_i
\newcommand{\CCdef}[1]{\xdef#1{\CCx}}     %defines #1 as the last C_i
\newcounter{cc}
\newcommand{\cc}{\stepcounter{cc}\ccx} %new constant c_i
\newcommand{\ccx}{c_{\arabic{cc}}}     %repeats the last c_i
\newcommand{\ccdef}[1]{\xdef#1{\ccx}}     %defines #1 as the last c_i

%\renewcommand\Re{\operatorname{Re}}
%\renewcommand\Im{\operatorname{Im}}

\newcommand\E{\operatorname{\mathbb E{}}}
\renewcommand\P{\operatorname{\mathbb P{}}}
\newcommand\Var{\operatorname{Var}}
\newcommand\Cov{\operatorname{Cov}}
\newcommand\Corr{\operatorname{Corr}}
\newcommand\Exp{\operatorname{Exp}}
\newcommand\Po{\operatorname{Po}}
\newcommand\Bi{\operatorname{Bi}}
\newcommand\Bin{\operatorname{Bin}}
\newcommand\Be{\operatorname{Be}}
\newcommand\Ge{\operatorname{Ge}}
\newcommand\NBi{\operatorname{NegBin}}
%\newcommand\PD{\operatorname{PD}}
\newcommand\Res{\operatorname{Res}}
\newcommand\fall[1]{^{\underline{#1}}}
\newcommand\rise[1]{^{\overline{#1}}}
%\newcommand\Int{\operatorname{Int}}
%\newcommand\arg{\operatorname{arg}}

\newcommand\ga{\alpha}
\newcommand\gb{\beta}
\newcommand\gd{\delta}
\newcommand\gD{\Delta}
\newcommand\gf{\varphi}
\newcommand\gam{\gamma}
\newcommand\gG{\Gamma}
\newcommand\gk{\varkappa}
\newcommand\gl{\lambda}
\newcommand\gL{\Lambda}
\newcommand\go{\omega}
\newcommand\gO{\Omega}
\newcommand\gs{\sigma}
\newcommand\gss{\sigma^2}
%\newcommand\gt{\tau}
\newcommand\gth{\theta}
\newcommand\gthh{\theta^2}
\newcommand\eps{\varepsilon}
\newcommand\ep{\varepsilon}
%\renewcommand\phi{\varphi}

\newcommand\bJ{\bar J}

\newcommand\cA{\mathcal A}
\newcommand\cB{\mathcal B}
\newcommand\cC{\mathcal C}
\newcommand\cD{\mathcal D}
\newcommand\cE{\mathcal E}
\newcommand\cF{\mathcal F}
\newcommand\cG{\mathcal G}
\newcommand\cH{\mathcal H}
\newcommand\cI{\mathcal I}
\newcommand\cJ{\mathcal J}
\newcommand\cK{\mathcal K}
\newcommand\cL{{\mathcal L}}
\newcommand\cM{\mathcal M}
\newcommand\cN{\mathcal N}
\newcommand\cO{\mathcal O}
\newcommand\cP{\mathcal P}
\newcommand\cQ{\mathcal Q}
\newcommand\cR{{\mathcal R}}
\newcommand\cS{{\mathcal S}}
\newcommand\cT{{\mathcal T}}
\newcommand\cU{{\mathcal U}}
\newcommand\cV{\mathcal V}
\newcommand\cW{\mathcal W}
\newcommand\cX{{\mathcal X}}
\newcommand\cY{{\mathcal Y}}
\newcommand\cZ{{\mathcal Z}}

\newcommand\tJ{{\tilde J}}


\newcommand\ett[1]{\boldsymbol1[#1]} 
%\newcommand\ett{\mathbf1} 
\newcommand\etta{\boldsymbol1} 
\def\[#1]{[\![#1]\!]}

\newcommand\smatrixx[1]{\left(\begin{smallmatrix}#1\end{smallmatrix}\right)}
%\newcommand\matrixx[1]{\left(\begin{matrix}#1\end{matrix}\right)}
%\newcommand\matrixx[1]{\begin{pmatrix}#1\end{pmatrix}}


\newcommand\limn{\lim_{n\to\infty}}
\newcommand\limN{\lim_{N\to\infty}}

\newcommand\qq{^{1/2}}
\newcommand\qqq{^{1/3}}
\newcommand\qqw{^{-1/2}}
\newcommand\qqqw{^{-1/3}}
\newcommand\qw{^{-1}}
\newcommand\qww{^{-2}}
%\newcommand\gsi{\sigma^{-1}}

\renewcommand{\=}{:=}

\newcommand\intoi{\int_0^1}
\newcommand\intoo{\int_0^\infty}
\newcommand\intoooo{\int_{-\infty}^\infty}
\newcommand\oi{[0,1]}
\newcommand\ooo{[0,\infty)}
\newcommand\oooo{(-\infty,\infty)}
\newcommand\setoi{\set{0,1}}

\newcommand\dtv{d_{\mathrm{TV}}}

\newcommand\dd{\,\textup{d}}

\newcommand{\pgf}{probability generating function}
\newcommand{\mgf}{moment generating function}
\newcommand{\chf}{characteristic function}
\newcommand{\ui}{uniformly integrable}
\newcommand\rv{random variable}
\newcommand\lhs{left-hand side}
\newcommand\rhs{right-hand side}

\newcommand\gnp{\ensuremath{G(n,p)}}
\newcommand\gnm{\ensuremath{G(n,m)}}
\newcommand\gnd{\ensuremath{G(n,d)}}
\newcommand\gnx[1]{\ensuremath{G(n,#1)}}
\newcommand\citetq[2]{\citeauthor{#2} \cite[{\frenchspacing #1}]{#2}} 

\newcommand\aam{{a_1,\dots,a_m}}
\newcommand\aamp{{a_1+\dots+a_m}}
\newcommand\maam{M_{\aam}}
\newcommand\mab{M_{a,b}}
\newcommand\ax{a^*}
%\newcommand\Ax{{N^*}}
%\newcommand\Ax{{N_*}}
\newcommand\Ax{N_*}
\newcommand\nux{^{(\nu)}}
\newcommand\ab{_{a,b}}
\newcommand\awb{_{a-1,b}}
\newcommand\eith{e^{\ii\gth}}
\newcommand\bfa{\mathbf{a}}
\newcommand\gfd[1]{\gf^{(#1)}}
\newcommand\A{B}
%\newcommand\{}
%\newcommand\{}
%\newcommand\{}
%\newcommand\{}
%\newcommand\{}
%QQQ

\newcommand{\Holder}{H\"older}
\newcommand\CS{Cauchy--Schwarz}
\newcommand\CSineq{\CS{} inequality}

\newcommand{\maple}{\texttt{Maple}}

\newcommand\ms[1]{\texttt{[ms #1]}}
%\newcommand{\ms}[1]{\par \vdots \texttt{[ms #1]} \par\vdots}
\newcommand\XXX{XXX \marginal{XXX}}
%\newcommand{\mer}[1]{\texttt{[#1]}\marginal{MER!}}
%\newcommand\REV{\marginal{Revidera!!}}
\newcommand\REM[1]{{\raggedright\texttt{[#1]}\par\marginal{XXX}}}
\newenvironment{OLD}{\Small \REM{Old stuff to be edited:}\par}{}
\newenvironment{comment}{\setbox0=\vbox\bgroup}{\egroup} %deletes!

\hyphenation{Upp-sala}

%\newcommand\urladdrx[1]{\urladdr{\def~{\~{}}#1}}
%\newcommand\urladdrx[1]{{\urladdr{\def~{{\tiny$\sim$}}#1}}}
\newcommand\urladdrx[1]{{\urladdr{\def~{\Tilde}#1}}}


%begin macros
%\def\L{{\cal L}}
%\baselineskip=14pt
%\parskip=10pt
%\def\halmos{\hbox{\vrule height0.15cm width0.01cm\vbox{\hrule height
%  0.01cm width0.2cm \vskip0.15cm \hrule height 0.01cm width0.2cm}\vrule
%  height0.15cm width 0.01cm}}
%\font\eightrm=cmr8 \font\sixrm=cmr6
%\font\eighttt=cmtt8
%\magnification=\magstephalf
%\def\N{{\cal N}}
%\def\P{{\cal P}}
%\def\Q{{\cal Q}}
%\def\R{{\cal R}}
%\def\1{{\overline{1}}}
%\def\2{{\overline{2}}}
%\renewcommand\Tilde{\char126\relax}
\renewcommand\Tilde{{\tiny$\sim$}}

\begin{document}
\title[The Mahonian probability distribution on words]
{The Mahonian Probability Distribution
on Words is Asymptotically Normal}
\thanks{Accompanied by Maple package MahonianStat available from
\hfill\break
\texttt{
http://www.math.rutgers.edu/\Tilde zeilberg/mamarim/mamarimhtml/mahon.html}.\\
The work of D. Zeilberger was supported in part
by the United States of America National Science Foundation.
The work of E. R. Canfield was supported in part by the NSA Mathematical
Sciences Program.
}


\date{Feb. 10, 2023} % (typeset \today{} \klockan)} %; revised ...

\author{Blair Seidler}
\address{ Department of Mathematics,
Rutgers University, NJ 08854, USA}
\email{blair [Dot] seidler [At] math [Dot] rutgers [Dot] edu}


\author{Svante Janson}
\address{Department of Mathematics, Uppsala University, PO Box 480,
SE-751~06 Uppsala, Sweden}
\email{svante.janson [At] math [Dot] uu [Dot] se}
\urladdrx{http://www.math.uu.se/~svante/}

\author{Doron ZEILBERGER}
\address{Mathematics Department, Rutgers University
(New Brunswick), Piscataway, NJ 08854, USA}
\email{zeilberg [At] math [Dot] rutgers [Dot] edu}



%\keywords{<keywords>}
\subjclass[2000]{} 
%{60C05 (68P10,68W40)} %%{Primary: <subject>; Secondary: <subject>}

\begin{comment}  % Some suggestions:
05 Combinatorics 
05C Graph theory [For applications of graphs, see 68R10, 90C35, 94C15]
05C05 Trees
05C07 Degree sequences
05C35 Extremal problems [See also 90C35]
05C40 Connectivity
05C65 Hypergraphs
05C80 Random graphs
05C90 Applications

60 Probability theory and stochastic processes
60C Combinatorial probability
60C05 Combinatorial probability

60F Limit theorems [See also 28Dxx, 60B12]
60F05 Central limit and other weak theorems
60F17 Functional limit theorems; invariance principles

60G Stochastic processes
60G09 Exchangeability
60G55 Point processes

60J Markov processes
60J65 Brownian motion [See also 58J65]
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)

60K Special processes
60K35 Interacting random processes; statistical mechanics type models;
      percolation theory [See also 82B43, 82C43] 

68 Computer science
68P Theory of data
68P05 Data structures
68P10 Searching and sorting

68Q Theory of computing
68Q25 Analysis of algorithms and problem complexity [See also 68W40] 

68R Discrete mathematics in relation to computer science
68R10 Graph theory [See also 05Cxx, 90B10, 90B35, 90C35]

68W Algorithms [For numerical algorithms, see 65-XX; for combinatorics and graph theory, see 68Rxx]
68W40 Analysis of algorithms [See also 68Q25]

82 Statistical mechanics, structure of matter
82B Equilibrium statistical mechanics
82B10 Quantum equilibrium statistical mechanics (general)
82B26 Phase transitions (general)

92 Biology and other natural sciences
92D Genetics and population dynamics
92D30 Epidemiology
\end{comment}

%\cite[Section X.6]{FellerI})


\begin{abstract}  This paper consists of two indpendent, but related parts.
In the first part we show how how to use symbolic computation to derive explicit expressions for the first few moments
of the number of implicants that a random Boolean function has, or equivalently
the number of fixed-dimensional subcubes contained in a random subset of the
$n$-deimensional cube. These explicit expressions suggest, but do not prove,
that these random variables are always asymptotically normal.

The second part is a full, human-generated proof, of this asymptotic normality.
\end{abstract}

%\dedicatory{}


\maketitle


\section{Introduction}\label{S:intro}




\section{Explicit Expressions for Moments of the Number of Low-Dimensional Subcubes in a Random Subest of the $n$-dimensional cube}\label{S:Expl}


COMING UP


\section{Asymptotic Normality }\label{S:ANor}

\def\nobibitem#1\par{}

\begin{thebibliography}{99}

%\bibitem[Andrews(1976)]{Andrews} 
%G. E. Andrews, 
%{\it The Theory of Partitions}, Addison-Wesley, 
%Reading, Mass., 1976.

\bibitem{FellerI}
W. Feller, 
\emph{An Introduction to Probability Theory and Its Application}, volume I, 
third edition, Wiley, %John Wiley \& Sons, 
New York, 1968.



\end{thebibliography}

\end{document}

