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%  Sample latex file
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%  Margaret Readdy
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%  Last edited: February 15, 2025
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\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{amstext,amsbsy}
\usepackage{mathrsfs}
\usepackage{fullpage}
\usepackage{epsfig,multicol}
\usepackage{amsthm}
\usepackage{amssymb,latexsym}
\usepackage{color}
\usepackage{tikz}
\usepackage{subfigure}
\usepackage{url}
\usepackage{enumerate}
\usetikzlibrary{calc}
\usetikzlibrary{arrows}




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\pdfpagewidth=8.5truein
\pdfpageheight=11truein


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\usepackage[colorlinks=true,citecolor=blue,urlcolor=blue]{hyperref}


\hypersetup{
colorlinks=true,
urlcolor=blue
}



\newtheorem{theorem}{Theorem}
\newtheorem{hypothesis}{Hypothesis}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{definition}{Definition}
\newtheorem{corollary}{Corollary}
\newtheorem{conjecture}{Conjecture}
\newtheorem{example}{Example}



\font\german = eufm10 scaled\magstep1
\font\Cp = msbm10



\newcommand{\vanish}[1]{}

\newcommand{\Sym}{\mathfrak S}
\newcommand{\Der}{\rm Der}

\setlength{\parskip}{2mm}


%%% Blackboard letters

\newcommand{\Ccc}{\hbox{\Cp C}}
\newcommand{\Fff}{\hbox{\Cp F}}
\newcommand{\Nnn}{\hbox{\Cp N}}
\newcommand{\Rrr}{\hbox{\Cp R}}
\newcommand{\Zzz}{\hbox{\Cp Z}}


\numberwithin{equation}{section}
%\usepackage[bindingoffset=0.2in,
%            left=1in,right=1in,top=1in,bottom=1in,
%            footskip=.25in]{geometry}%Sets margins
%\pagenumbering{gobble}%No page numbers



\newcommand{\timesdots}{\times \cdots \times}
\newcommand{\coveredby}{\prec}


\DeclareMathOperator{\des}{des}


\begin{document}



\title{\sc Math 391 Sample LaTeX Document}


\author{\sc Margaret Readdy}


%%%\date{}
\date{\today}

\maketitle



\begin{abstract}
This is a sample latex document.
\end{abstract}







\section{Useful LaTeX help}



See
\href{http://mathlab.cit.cornell.edu/support/latex/sample/sample.pdf}
{mathlab.cit.cornell.edu/support/latex/sample/sample.pdf}.


To run latex, type
$$
    \mbox{\tt pdflatex filename.tex}
$$  
This will generate a pdf of your file named {\tt filename.pdf}.
Sometimes you will need to latex twice to update pointers in the file.



\section{Some math}


Here is some math:

A permutation
 $\pi = \pi_1 \cdots \pi_n$ in the symmetric group  $\Sym_n$ is
a {\em derangement} if it has no fixed points, that is, 
$\pi_i \neq i$ for $i = 1, \ldots, n$.



See Figure~\ref{figure_Fibonacci} for a small table.


\begin{figure}
{\small
\begin{eqnarray*}
  F_0 = 1\\
  F_1 = 1\\
  F_2 = 2\\
  F_3 = 3\\
  F_4 = 5
\end{eqnarray*}
}
\caption{Some Fibonacci numbers.}
\label{figure_Fibonacci}
\end{figure}


\begin{conjecture}
Here is a guess.
\end{conjecture}



\begin{lemma}
Differentiability of $f(x)$ implies continuity.
\end{lemma}
\begin{proof}
Here is where you can type a proof.
\end{proof}


Recall that Euler's theorem states
$$
   a^{\varphi(b)} \equiv 1 \mod{b}
$$
whenever $\gcd(a,b) = 1$.




\section{Gilbreath's Principle}
\label{chapter_introduction}


Like many magic tricks,
we begin with a standard deck of cards.
Each card has two ``decorations'',
namely, a {\em denomination} from the set
$$
     \{A, 2, 3, \ldots, 9, 10, J, Q, K\},
$$
where $A$ indicates an Ace,
$J$ a Jack,
$Q$ a Queen
and 
$K$ a King,
and a {\em suit}
from the set
$$
     \{\clubsuit, \diamondsuit, \heartsuit, \spadesuit\},
$$
where 
$\clubsuit$ is a club,
$\diamondsuit$ is a diamond,
$\heartsuit$ is a heart,
and
$\spadesuit$ is a spade.
There are $13$ denominations and $4$ suits, so the total number of
cards in a standard deck is $13 \cdot 4 = 52$ cards.


A {\em riffle shuffle}, or simply,
a {\em shuffle},  is the result of splitting a deck cards into two piles and taking cards randomly from each pile to form a new pile.  As an example, if we have a deck of $5$ cards $\{A,B,C,D,E\}$ split into piles
$ABC$ and $DE$, there are $10$ possible shuffles of 
these two piles, namely 
\begin{eqnarray*}
     ABCDE, DABCE, ADBCE, ABDCE,
     DEABC, \\
     DAEBC, 
     DABEC, ADEBC, ADBEC, ABDEC.
\end{eqnarray*}


To find out about Gilbreath's Principle, see~\cite{Gilbreath}.


\newpage

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\newcommand{\web}[3]{{\sc #1,} #2, #3.}
\newcommand{\JCTA}{J.\ Combin.\ Theory Ser.\ A}
\newcommand{\AdvancesinMathematics}{Adv.\ Math.}
\newcommand{\JournalofAlgebraicCombinatorics}{J.\ Algebraic Combin.}

\newcommand{\appear}[3]{{\sc #1,} #2, to appear in {\it #3}}

{\small

\begin{thebibliography}{99}


\bibitem{Gilbreath}

\article{Norman Gilbreath}
        {Magnetic Colors Principle}
        {The Linking Ring}
        {38}{1958}{60}

\end{thebibliography}


\end{document}
