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\pagestyle{fancy} \lhead{{\sf Homework 2 $|$ Due Wednesday September 17 ()
}} \rhead{\thepage} \cfoot{{\sf Math 4110 $|$ Combinatorics
    $|$ Fall 2025 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\textbf{Topics}: Elementary counting problems, the binomial
theorem, combinatorial reciprocity.

\begin{enumerate}

\item Without the use of Google or AI, give a combinatorial proof of
  the following identity:
  \[
  k\binom{n}{k}=n\binom{n-1}{k-1}.
  \]
  Specifically, come up with a problem whose enumeration is
  the LHS and the RHS. \medskip


\item If $r\in\R$ and $k\in\N_0$, then the generalized binomial
  coefficient is defined as
  \[
  \binom{r}{k}=\frac{n(n-1)\cdots(n-k+1)}{k!}.
  \]
  Using generating functions, this leads to the well-known
  combinatorial reciprocity formula
  \[
  \multiset{n}{k}=\binom{-n}{k}(-1)^k.
  \]
  Now, suppose that $n\in\N_0$ is fixed, but $k$ is allowed to be
  negative. Define
  \[
  \binom{n}{-k}=\frac{n(n+1)\cdots(n+k-1)}{k!}.
  \]
  Does this lead to an analogous combinatorial reciprocity formula?
  Explore what happens if $n$ is allowed to be negative, and then for
  arbitrary $n\in\R$. \\


\item Recall \emph{Vandermode's identity}, which says that
\[
\binom{n+m}{k}=\sum_{i=0}^k\binom{n}{i}\binom{m}{k-1}.
\]
Explore this if one or more of the parameters are negative. Find one or more combinatorial reciprocity identities, and explain what it means. Does an analogous proof with generating functions work? Show a few small examples to illustrate this.

  
\end{enumerate}

\end{document}


%% 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}
  
  

