% Like most advanced LaTeX files, this one begins with a lot of
% boilerplate. You don't need to understand (or even read) most of it.
% All you need to do is fill in your name, UMN ID, email address,
% and the number of the pset. (Search for "METADATA" to find the place
% for this.) Then, you can go straight to the "EXERCISE 1"
% section and start writing your solutions.
% The "VARIOUS USEFUL COMMANDS" section is probably worth taking a
% look at at some point.

%----------------------------------------------------------------------------------------
%	PACKAGES AND OTHER DOCUMENT CONFIGURATIONS
%----------------------------------------------------------------------------------------
\documentclass[paper=a4, fontsize=12pt]{scrartcl} % A4 paper and 12pt font size
\usepackage[T1]{fontenc} % Use 8-bit encoding that has 256 glyphs
\usepackage[english]{babel} % English language/hyphenation
\usepackage{amsmath,amsfonts,amsthm,amssymb} % Math packages
\usepackage{mathrsfs}    % More math packages
\usepackage{sectsty}  % Allows customizing section commands
\allsectionsfont{\centering \normalfont\scshape} % Make all section titles centered, the default font and small caps %remove this to left align section tites
\usepackage{hyperref} % Turns cross-references into hyperlinks,
                      % and defines \url and \href commands.
\usepackage{graphicx} % For embedding graphics files.
\usepackage{framed}   % For the "leftbar" environment used below.
\usepackage{ifthen}   % Used for the \powset command below.
\usepackage{lastpage} % for counting the number of pages
\usepackage[headsepline,footsepline,manualmark]{scrlayer-scrpage}
\usepackage[height=10in,a4paper,hmargin={1in,0.8in}]{geometry}
\usepackage[usenames,dvipsnames]{xcolor}
\usepackage{tikz}     % This is a powerful tool to draw vector
                      % graphics inside LaTeX. In particular, you can
                      % use it to draw graphs.
\usepackage{verbatim} % For the "verbatim" environment, in which
                      % special symbols can be used freely without
                      % confusing the compiler. (And it's typeset in
                      % a constant-width font.)
                      % Useful, e.g., for quoting code (or ASCII art).
\usepackage{ytableau}

%\numberwithin{table}{section} % Number tables within sections (i.e. 1.1, 1.2, 2.1, 2.2 instead of 1, 2, 3, 4)

\setlength\parindent{20pt} % Makes indentation for paragraphs longer.
                           % This makes paragraphs stand out more.

%----------------------------------------------------------------------------------------
%	VARIOUS USEFUL COMMANDS
%----------------------------------------------------------------------------------------
% The commands below might be convenient. For example, you probably
% prefer to write $\powset[2]{V}$ for the set of $2$-element subsets
% of $V$, rather than writing $\mathcal{P}_2(V)$.
% Notice that you can easily define your own commands like this.
% Caveat: Some of these commands need to be properly "guarded" when
% they occur in subscripts or superscripts. So you should not write
% $K_\CC$, but rather $K_{\CC}$.
\newcommand{\CC}{\mathbb{C}} % complex numbers
\newcommand{\RR}{\mathbb{R}} % real numbers
\newcommand{\QQ}{\mathbb{Q}} % rational numbers
\newcommand{\NN}{\mathbb{N}} % nonnegative integers
\newcommand{\Z}[1]{\mathbb{Z}/#1\mathbb{Z}} % integers modulo k
                                            % (syntax: "\Z{k}")
\newcommand{\ZZ}{\mathbb{Z}} % integers
\newcommand{\id}{\operatorname{id}} % identity map
\newcommand{\lcm}{\operatorname{lcm}}
% Lowest common multiple. For historical reasons, LaTeX has a \gcd
% command built in, but not an \lcm command. The preceding line
% rectifies that.
\newcommand{\rev}{\operatorname{rev}} % reversal of a walk
\newcommand{\powset}[2][]{\ifthenelse{\equal{#2}{}}{\mathcal{P}\left(#1\right)}{\mathcal{P}_{#1}\left(#2\right)}}
% $\powset[k]{S}$ stands for the set of all $k$-element subsets of
% $S$. The argument $k$ is optional, and if not provided, the result
% is the whole powerset of $S$.
\newcommand{\set}[1]{\left\{ #1 \right\}}
% $\set{...}$ compiles to {...} (set-brackets).
\newcommand{\abs}[1]{\left| #1 \right|}
% $\abs{...}$ compiles to |...| (absolute value, or size of a set).
\newcommand{\tup}[1]{\left( #1 \right)}
% $\tup{...}$ compiles to (...) (parentheses, or tuple-brackets).
\newcommand{\ive}[1]{\left[ #1 \right]}
% $\ive{...}$ compiles to [...] (Iverson bracket, aka truth value).
\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}
% $\floor{...}$ compiles to |_..._| (floor function).
\newcommand{\verts}[1]{\operatorname{V}\left( #1 \right)}
% $\verts{...}$ compiles to V(...) (vertex set of a graph/digraph).
\newcommand{\edges}[1]{\operatorname{E}\left( #1 \right)}
% $\edges{...}$ compiles to E(...) (edge set of a graph).
\newcommand{\arcs}[1]{\operatorname{A}\left( #1 \right)}
% $\arcs{...}$ compiles to A(...) (arc set of a digraph).
\newcommand{\lf}[2]{#1^{\underline{#2}}}
% $\lf{...1}{...2}$ compiles to $...1^{\underline{...2}}$.
% This is a notation for the falling factorial.
\newcommand{\underbrack}[2]{\underbrace{#1}_{\substack{#2}}}
% $\underbrack{...1}{...2}$ yields
% $\underbrace{...1}_{\substack{...2}}$. This is useful for doing
% local rewriting transformations on mathematical expressions with
% justifications. For example, try this out:
% $ \underbrack{(a+b)^2}{= a^2 + 2ab + b^2 \\ \text{(by the binomial formula)}} $
\newcommand{\horrule}[1]{\rule{\linewidth}{#1}} % Create horizontal rule command with 1 argument of height
\newcommand{\nnn}{\nonumber\\} % Don't number this line in an "align" environment, and move on to the next line.

%----------------------------------------------------------------------------------------
%	MAKING SUMMATION SIGNS ALWAYS PUT THEIR BOUNDS ABOVE AND BELOW
%	THE SIGN
%----------------------------------------------------------------------------------------
% The following are hacks to ensure that sums (such as
% $\sum_{k=1}^n k$) always put their bounds (i.e., the $k=1$ and the
% $n$) underneath and above the sign, as opposed to on its right.
% Same for products (\prod), set unions (\bigcup) and set
% intersections (\bigcap). Remove the 8 lines below if you do not want
% this behavior.
\let\sumnonlimits\sum
\let\prodnonlimits\prod
\let\cupnonlimits\bigcup
\let\capnonlimits\bigcap
\renewcommand{\sum}{\sumnonlimits\limits}
\renewcommand{\prod}{\prodnonlimits\limits}
\renewcommand{\bigcup}{\cupnonlimits\limits}
\renewcommand{\bigcap}{\capnonlimits\limits}

%----------------------------------------------------------------------------------------
%	ENVIRONMENTS
%----------------------------------------------------------------------------------------
% The incantations below define how theorem environments
% (\begin{theorem} ... \end{theorem}) and their likes will look like.
\newtheoremstyle{plainsl}% <name>
  {8pt plus 2pt minus 4pt}% <Space above>
  {8pt plus 2pt minus 4pt}% <Space below>
  {\slshape}% <Body font>
  {0pt}% <Indent amount>
  {\bfseries}% <Theorem head font>
  {.}% <Punctuation after theorem head>
  {5pt plus 1pt minus 1pt}% <Space after theorem headi>
  {}% <Theorem head spec (can be left empty, meaning `normal')>

% Environments which make the text inside them slanted:
\theoremstyle{plainsl}
  \newtheorem{theorem}{Theorem}[section]
  \newtheorem{proposition}[theorem]{Proposition}
  \newtheorem{lemma}[theorem]{Lemma}
  \newtheorem{corollary}[theorem]{Corollary}
  \newtheorem{conjecture}[theorem]{Conjecture}
% Environments that don't:
\theoremstyle{definition}
  \newtheorem{definition}[theorem]{Definition}
  \newtheorem{example}[theorem]{Example}
  \newtheorem{exercise}[theorem]{Exercise}
  \newtheorem{examples}[theorem]{Examples}
  \newtheorem{algorithm}[theorem]{Algorithm}
  \newtheorem{question}[theorem]{Question}
 \theoremstyle{remark}
  \newtheorem{remark}[theorem]{Remark}
\newenvironment{statement}{\begin{quote}}{\end{quote}}
\newenvironment{fineprint}{\begin{small}}{\end{small}}

%----------------------------------------------------------------------------------------
%	METADATA
%----------------------------------------------------------------------------------------
\newcommand{\myname}{Darij Grinberg} % ENTER YOUR NAME HERE
\newcommand{\myid}{00000000} % ENTER YOUR UMN ID HERE
\newcommand{\mymail}{dgrinber@umn.edu} % ENTER YOUR EMAIL HERE
\newcommand{\psetnumber}{2} % ENTER THE NUMBER OF THIS PSET HERE

%----------------------------------------------------------------------------------------
%	HEADER AND FOOTER
%----------------------------------------------------------------------------------------
\ihead{Solutions to homework set \#\psetnumber} % Page header left
\ohead{page \thepage\ of \pageref{LastPage}} % Page header right
\ifoot{\myname, \myid} % left footer
\ofoot{\mymail} % right footer

%----------------------------------------------------------------------------------------
%	TITLE SECTION
%----------------------------------------------------------------------------------------
\title{	
\normalfont \normalsize 
\textsc{University of Minnesota, School of Mathematics} \\ [25pt] % Your university, school and/or department name(s)
\horrule{0.5pt} \\[0.4cm] % Thin top horizontal rule
\huge Math 5705: Enumerative Combinatorics, \\
Fall 2018:
Homework \psetnumber\\% The assignment title
\horrule{2pt} \\[0.5cm] % Thick bottom horizontal rule
}
\author{\myname}

\begin{document}

\maketitle % Print the title

\begin{center} % Delete this if you want to save space!
{\large due date: \textbf{Wednesday, 26 September 2018} at the beginning of class, \\
or before that by email or canvas.

Please solve \textbf{at most 3 of the 5 exercises}!}
\end{center}

%----------------------------------------------------------------------------------------
%	EXERCISE 1
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 1}

\subsection{Problem}

For any nonnegative integers $a$ and $b$ and any real $x$,
prove that
\begin{equation}
\dbinom{x}{a} \dbinom{x}{b}
= \sum_{r = \max\set{a, b}}^{a+b}
    \dbinom{a}{a+b-r} \dbinom{r}{a} \dbinom{x}{r} .
\label{eq.exe.xaxb.claim}
\end{equation}

\subsection{Solution}

[...]

%----------------------------------------------------------------------------------------
%	EXERCISE 2
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 2}

\subsection{Problem}

Let $n \in \NN$ and $k \in \NN$. Prove that
\begin{equation}
\sum_{i=0}^n \dbinom{n}{i} \dbinom{n-i}{k-2i} 2^{k-2i} = \dbinom{2n}{k} .
\label{eq.exe.n-pairs-of-shoes.claim}
\end{equation}

[\textbf{Hint:} You have $n$ pairs of shoes
$\tup{L_1, R_1}, \tup{L_2, R_2}, \ldots, \tup{L_n, R_n}$,
where the $2n$ shoes $L_1, R_1, L_2, R_2, \ldots, L_n, R_n$
are all distinguishable.
You grab $k$ of these $2n$ shoes at random (i.e., pick
a $k$-element subset of the set of all $2n$ shoes).
For a given $i \in \set{0, 1, \ldots, n}$, what is the
probability that among your $k$ shoes are exactly $i$
pairs?]

\subsection{Solution}

[...]

%----------------------------------------------------------------------------------------
%	EXERCISE 3
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 3}

\subsection{Problem}

Let $n \in \NN$.
For each $i \in \set{0, 1, 2}$, we let $g_{n, i}$ denote the
number of all subsets $S$ of $\ive{n}$ satisfying
$\abs{S} \equiv i \mod 3$.

\begin{enumerate}

\item[(a)]
Show that if $n > 0$, then
\[
g_{n, 0} = g_{n-1, 0} + g_{n-1, 2}; \qquad
g_{n, 1} = g_{n-1, 1} + g_{n-1, 0}; \qquad
g_{n, 2} = g_{n-1, 2} + g_{n-1, 1}.
\]

\item[(b)]
Find closed-form expressions (with no summation signs) for
$g_{n, 0}, g_{n, 1}, g_{n, 2}$ depending on the
remainder of $n$ upon division by $3$.

\end{enumerate}

\subsection{Remark}

\begin{remark}
The combinatorial interpretation of binomial coefficients shows
that
\[
g_{n, i} = \sum_{\substack{k \in \ZZ; \\ k \equiv i \mod 3}} \dbinom{n}{k}
\qquad \text{for each $i$}.
\]
This is not what the problem is asking for -- find formulas
with no summation signs.
\end{remark}

\subsection{Solution}

[...]

%----------------------------------------------------------------------------------------
%	EXERCISE 4
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 4}

\subsection{Problem}

Let $n \in \NN$ be positive.
Let $m \in \NN$.
Prove that
\begin{equation}
\sum_{k=0}^m \tup{-1}^k \dbinom{n}{k} = \tup{-1}^m \dbinom{n-1}{m} .
\label{eq.exe.binom.altsum-partial.claim}
\end{equation}

\subsection{Solution}

[...]

%----------------------------------------------------------------------------------------
%	EXERCISE 5
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 5}

\subsection{Problem}

Let $n \in \NN$.
If $\mathbf{i} = \tup{i_1, i_2, \ldots, i_n} \in \set{0, 1}^n$
and $k \in \ive{n}$,
then
\begin{itemize}
\item we say that $k$ is a \textit{1-position} of $\mathbf{i}$
      if $i_k = 1$;
\item we say that $k$ is a \textit{10-position} of $\mathbf{i}$
      if $k < n$, $i_k = 1$ and $i_{k+1} = 0$.
\end{itemize}

For example, the $7$-tuple $\tup{0, 1, 1, 0, 1, 0, 1}$ has
1-positions $2, 3, 5, 7$ and 10-positions $3, 5$.

It is easy to see that for each $k \in \ZZ$, the number
of all $n$-tuples $\mathbf{i} \in \set{0, 1}^n$ having
exactly $k$ 1-positions is $\dbinom{n}{k}$.
(In fact, these $n$-tuples are in bijection with the
$k$-element subsets of $\ive{n}$.)
In this problem, we shall count the $n$-tuples having
exactly $k$ 10-positions.

We use the notation $a \% b$ for the remainder of an integer
$a$ upon division by a positive integer $b$.
For example, $5 \% 3 = 2$.
Also, $\floor{x}$ denotes the integer part (i.e., floor)
of a real number $x$ (that is, the largest integer that
is smaller or equal to $x$).

Let $A : \set{0, 1}^n \to \set{0, 1}^n$ be the map that
sends any $n$-tuple $\tup{i_1, i_2, \ldots, i_n}$ to the
$n$-tuple $\tup{j_1, j_2, \ldots, j_n}$, where
\[
j_k = \tup{i_1 + i_2 + \cdots + i_k} \% 2
\qquad \text{for all $k$.}
\]
For example,
$A \tup{ \tup{0, 1, 1, 0, 0, 1, 0} }
 =       \tup{0, 1, 0, 0, 0, 1, 1}$.

Prove the following:

\begin{enumerate}

\item[(a)] The map $A$ is bijective.

\item[(b)] If the number of 1-positions of some
$n$-tuple $\mathbf{i} \in \set{0, 1}^n$ is $p$,
then the number of 10-positions of
the $n$-tuple $A \tup{ \mathbf{i} }$ is $\floor{p/2}$.

\item[(c)] Let $k \in \ZZ$.
Then, the number of $n$-tuples $\mathbf{i} \in \set{0, 1}^n$
having exactly $k$ 10-positions is
$\dbinom{n+1}{2k+1}$.

\end{enumerate}

\subsection{Solution}

[...]

\end{document}

