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\pagestyle{fancy} \lhead{{\sf Homework 6 $|$ Due October 24 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4110 $|$ Combinatorics
    $|$ Fall 2025 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\textbf{Topics}: Eulerian and Mahonian statistics


\begin{enumerate}


\item In class, we computed the descent, excedance, weak excedance, fixed points, inversion, and major index, for the $6$ permutations in $\Sym_3$. They are summarized by the following table and polynnomials..
\[
\begin{tabular}{|cc|cccc|cc|} \hline
  $w$ & $\widehat{w}$ & $\exc(\pi)$ & $\wexc(\pi)$ & $\fix(\pi)$ & $\des(\pi)$ & $\inv(\pi)$ & $\maj(\pi)$ \\ \hline
$(1)(2)(3)$ & 123           & 0 & 3 & 3 & 0 & 0 & 0 \\
$(1)(3\u{2})$ & 1\u{3}2     & 1 & 2 & 1 & 1 & 1 & 2 \\
$(2\u{1})(3)$ & \u{2}13     & 1 & 2 & 1 & 1 & 1 & 1 \\
$(3\u{1}\u{2})$ & 2\u{3}1   & 2 & 2 & 0 & 1 & 2 & 2 \\
$(32\u{1})$ & \u{3}12       & 1 & 1 & 0 & 1 & 2 & 1 \\
$(3\u{1})(2)$ & \u{3}\u{2}1 & 1 & 2 & 1 & 2 & 3 & 3 \\ \hline
\end{tabular}
\]
\[
\sum_{w\in\Sym_3}x^{\exc(w)}=\sum_{w\in\Sym_3}x^{\des(w)}=1+4x+4x^2,\qquad
\sum_{w\in\Sym_3}x^{\inv(w)}=\sum_{w\in\Sym_3}x^{\maj(w)}=1+2x+2x^2+x^3.
\]
Create a similar table for the $24$ permuations in $\Sym_4$, and write the corresponding polynomials. You may use ChatGPT to help as long as you summarize your steps and usage.

\item In class, we saw the 1978 bijection $\varphi\colon
  \Sym_n\to\Sym_n$ of Foata and Sch\"{u}tzenberger for which
  $\maj(\pi)=\varphi(\inv(\pi))$. This is defined by starting with a word $w=w_1\cdots w_n$, and recursively defining the words $\gamma_1,\dots,\gamma_n$, where each $\gamma_i$ has length $i$, according to the following rules:
 %%
\begin{enumerate}
    \item Set $\gamma_1=w_1$.
    \item Given $\gamma_i$:
    \begin{enumerate}
        \item Split it into subwords:
        \begin{enumerate}
            \item If $w_i>w_{i+1}$ split after each number $>w_{i+1}$
            \item If $w_i>w_{i+1}$ split after each number $<w_i$.
        \end{enumerate}
        \item Cyclically shift each resulting compartment $1$ to the right
        \item Add $w_{i+1}$ at the end to get $\gamma_{i+1}$
    \end{enumerate}
    \item The image of $w$ is $\gamma_n$.
\end{enumerate}
In class, we did an example to demonstrate that $\varphi(683941725)=364891725$, and we arranged each step in a row in a table, like the following:
\[
\begin{tabular}{|lc|lcclll|} \hline
  unsplit & $\inv(\gamma_i)$ && $w_i$ vs.\ $w_{i+1}$ && split && cyclically shift  \\ \hline
$\gamma_1=6$ & 0 && $6\;\Red{<8}$ && 6 && 6 \\
  $\gamma_2=68$ & 0 && $8\;\Red{>3}$ && $6\mid 8$ && $6\mid 8$ \\
    \hspace{6mm}\vdots & \vdots && \vdots && \hspace{3mm}\vdots && \hspace{3mm}\vdots  \\
$\gamma_8=63894712$ & 18 && $2\;\Red{<5}$ && $\Blue{63}\mid \Blue{894}\mid \Blue{71}\mid 2$ && $\Blue{36}\mid \Blue{489}\mid \Blue{17}\mid 2$ \\
$\gamma_9=364891725$ & 18 && -- && -- && -- \\ \hline
\end{tabular}
\]
Carry out a similar exercise for the permutation $w=469528137$.



\item Recall the \emph{fundamental bijection} $w\mapsto\hat{w}$ of Foata that removes the parantheses from a permutation in standard cycle notation.
  In class, we considered the following permutation under the inverse map:
\[
\renewcommand{\arraystretch}{.8}
\setlength{\arraycolsep}{2pt}
\widehat{w}=
\begin{bmatrix}
\Red{1} & 2 & \Red{3} & 4 & \Red{5} & \Red{6} & \Red{7} & 8 \\
2 & 1 & 6 & 3 & 5 & 7 & 8 & 4
\end{bmatrix}
\longmapsto(2\u{1})(6\u{35})(7)(8\u{4})=w,
\]
where the (strict) excedences in $\hat{w}$ are underlined, and the weak excedences in $w$ are in red. Recall that these are defined as
\[
\Wexc(w)=\{i\in[n]:w(i)\geq i\},\qquad
\Exc(w)=\{i\in[n]:w(i)>i\},
\]
and note that $1$ will \emph{always} be a weak excedence. Show how to construct a bijection 
\[
\Wexc(\hat{w})\setminus\{1\}\;\longmapsto\;\Exc(w).
\]
You are encouraged to use ChatGPT, but you must document your usage, and note that it seems to be \emph{very} prone to making errors on this particular problem! \\
  

  
\end{enumerate}

\end{document}






  




\item In class, we showed that the \emph{inversion table}
  \[
  \Sym_n\longto [0,n-1]\times[0,n-2]\times\cdots\times[0,0],\qquad
  w\longmapsto I(w)
  \]
  is a bijection, from which we concluded that  
  \[
  \sum_{\pi\in\Sym_n}x^{\inv(\pi)}=(1+x)(1+x+x^2)\cdots(1+x+x^2+\cdots x^{n-1}). 
  \]
  Then, we constructed a bijection $\Sym_n\to\Sym_n$ that sent $\inv$
  to $\maj$, from which it follows that $\maj$ has the same generating
  function. Given an alternate direct proof of this result. You are encouraged to use ChatGPT for guidance, but you must write up the solution on your own. (In my experience, the answer is correct, but lacking in details.) You must include enough details so that your write-up can be understood by a classmate. Include an example to illustrate it. 
  

