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\pagestyle{fancy} \lhead{{\sf Homework 3 $|$ Due September 24 (Wednesday)
}} \rhead{\thepage} \cfoot{{\sf Math 4110 $|$ Combinatorics
    $|$ Fall 2025 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\textbf{Topics}: Lattice paths, generating functions, combinatorial reciprocity.

\begin{enumerate}

  
\item Consider the set of \emph{lattice paths} in $\Z\times\Z$ that
  only take \emph{positive steps}, i.e., $(1,0)$ or $(0,1)$, from the
  origin $(0,0)$ to $(m,n)$, for some $m\geq n$. Two examples of these
  are shown below for $m=12$ and $n=8$.
  \[
  \begin{tikzpicture}[scale=.5]
    \def\xmax{12}
    \def\ymax{8}
    \tikzstyle{p} = [circle, draw, line width=.5pt, fill=black,inner sep=0pt, minimum size=1mm]
    \node[p] at (0,0) {};
    \node[p] at (\xmax,\ymax) {};
    \draw[thick] (0,-.5) to (0,\ymax+.5);
    \draw[thick] (-.5,0) to (\xmax+.5,0);
    \foreach \x in {1,...,\xmax} {
      \draw[dotted] (\x,-.5) to (\x,\ymax+.5);
    }
    \foreach \y in {1,...,\ymax} {
      \draw[dotted] (-.5,\y) to (\xmax+.5,\y);
    }
    \draw[dashed] (0,0) to (\ymax+.5,\ymax+.5);
    \draw[very thick,blue] (0,0)--(3,0)--(3,2)--(5,2)--(5,3)--(9,3)--(9,8)--(12,8);
    \draw[very thick,red] (0,0)--(0,3)--(3,3)--(3,5)--(6,5)--(6,7)--(12,7)--(12,8);
  \end{tikzpicture}
  \]
  
  \begin{enumerate}
  \item Show that there are exactly $\binom{n+m}{n}$ such lattice paths.

  \item Say that a lattice path is \emph{good} if it stays below the diagonal, except possibly at its starting and ending points. All other paths are \emph{bad}. Show that there is a bijection between the sets of
    \begin{itemize}
    \item bad lattice paths from $(1,0)$ to $(m,n)$,
    \item all lattice paths from $(0,1)$ to $(m,n)$. 
    \end{itemize}
    Then show that these sets have size $\binom{n+m-1}{m}$. 

  \item Subtract the answer from Part~(b) from that of Part~(a), and
    simplify, to get that there are
    \[
    G(m,n):=\frac{m-n+1}{m+1}\binom{m+n}{n}
    \]
    good lattice paths from $(0,0)$ to $(m,n)$.

  \item Derive a reciprocity formula involving the number of path the
    touch, but don't cross, the diagonal.
    
  \end{enumerate}
  
  
%% Will be moved to HW 3
\item In class, we proved the following identity:
  \[
  \frac{1}{\sqrt{1-4x}}=\sum_{n\geq 0}\binom{2n}{n}x^n.
  \]
  \begin{enumerate}
    \item Use this to prove a related identity:
    \[
    \sum_{k=0}^n\binom{2k}{k}\binom{2(n-k)}{n-k}=4^n.
    \]
    You are encouraged to use tools like ChatGPT for help; just document
    its usage.
    
  \item ChatGPT says about the above identity: ``\emph{If you prefer a combinatorial view: the left side counts ordered pairs of Dyck-path-like objects whose total size is $n$.}'' Use tools like ChatGPT and/or Google to dive into this and explain exactly what this means. Include several examples. \\
  \end{enumerate}

  
\item In class, we proved that
  \[
  x^n=\sum_{k=0}^nS(n,k)(x)_k,
  \]
  where $S(n,k)$ are the Stirling numbers of the second kind, which
  count the number of partitions of $[n]$ into $k$ blocks. Then, we
  did an explicit example to demonstrate this using $n=3$. Carry out
  the details of this for $n=4$.
  
  % Here's a good environment to use for this:
  %
  %  \begin{align*}
  %    x^4 &= \\
  %        &= \\
  %        &=    
  %  \end{align*}
  
\end{enumerate}

\end{document}

\item 



