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\pagestyle{fancy} \lhead{{\sf Homework 5 $|$ Due October 10 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4110 $|$ Combinatorics
    $|$ Fall 2025 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\textbf{Topics}: Permutation statistics


\begin{enumerate}



\item Recall that $\stirlingI{n}{k}=c(n,k)$ is the number of permutations in $\Sym_n$ with $k$ cycles. In class, we proved that
  \[
  \sum_{\pi\in\Sym_n}x^{\cyc(\pi)}=\sum_{k=0}^n\stirlingI{n}{k}x^k
  =x(x+1)(x+2)\cdots(x+n-1)=x^{\overline{n}}.
  \]
  In this problem, we will explore this for $n=5$.
  \begin{enumerate}
    \item Determine how many cycle types there are for in $S_5$, and
      the size of each one using elementary counting arguments.
    \item Carry out the details to verify the identity above for $n=5$. 
  \end{enumerate}
  
  
%%
\item Recall the \emph{fundamental bijection}
  \[
  \theta\colon\Sym_n\longto\Sym_n,\qquad\theta\colon\pi\longmapsto\widehat{\pi},
  \]
  that maps a permutation in its standard cycle notation to the
  permutation whose 1-line notation is the result of removing the
  parantheses.
  \begin{enumerate}
  \item Prove that the following are equivalent:
    \begin{enumerate}[(i)]
    \item $\pi$ is a product of cycles of length at most $2$,
    \item $\pi^{-1}=\pi$.
    \end{enumerate}
    \item Prove that if $\widehat{\pi}=\pi$, then $\pi^{-1}=\pi$.
  \item Prove that the number of such permutations is a Fibonacci
    number. [\emph{Hint}: show that they satisfy the Fibonacci
      recurrence.]
  \end{enumerate}

  

\item In class, we proved the identity
  \[
  \sum_{\pi\in S_n}x^{\inv(\pi)}=(1+x)(1+x+x^2)\cdots(1+x+x^2+\cdots+x^{n-1}).
  \]
  Carry out the details of this for $n=4$. \\


\item Compute the inversion table of the permutation $w=854192637$ in
  $\Sym_9$. Then, by inserting the numbers $9,8,\dots,1$ sequentially, construct the permutation $w\in\Sym_9$ that has inversion table $I(w)=(6,3,0,2,4,1,1,1,0)$. \\
  
\item In class, we proved that the \emph{Eulerian polynomials} satisfy
  \[
  A_n(x)=\sum_{k=0}^nA(n,k)x^k=\sum_{\pi\in\Sym_n}x^{\des(\pi)}.
  \]
  Carry out the details of this for $n=4$. That is, for each of the
  $4!=24$ permutations $\pi\in\Sym_4$, write $\des(\pi)$, and use this
  to find the polynomial $A_4(x)$.
  

\end{enumerate}

\end{document}
