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\begin{document}

\begin{center}
\textbf{Fall 2021, Math 596:\ Week 8 Problem Set} \\
\textbf{Due:\ Thursday, October 21st, 2021} \\
\textbf{Face Lattice Combinatorics}
\end{center}

\begin{center}
\begin{tabular}{r@{\,\,\,}p{4in}}
Name: & 
% INSERT NAME HERE
X
\\
Collaborators: & 
% LIST COLLABORATORS HERE
X, X, X
\end{tabular}
\end{center}


\bigskip
\noindent
\textbf{Homework problems.}
You must submit \emph{all} homework problems in order to receive full credit.  

\begin{enumerate}[(H1)]

\item 
In the assignment PDF is a picture of the 3-dimensional associahedron $A_5$, and the final page depicts the face lattice of $A_5$ (except for the empty face), where each face is represented by a collection of non-crossing diagonals of a hexagon.  Your goal for this problem is to connect these depictions and explore the structure of the face lattice.  Your answer should be a paragraph or two addressing the following (plus any other patterns you notice).  

\begin{itemize}
\item 
When is one collection of non-crossing diagonals ``above'' another in the face lattice?  

\item 
Given a collection of non-crossing diagonals, what determines the dimension of the corresponding face?  

\item 
What determines which facets are square vs pentagonal?  In higher dimensional associahedra, what will the different ``types'' of facets be?  

\item 
Each row of the face lattice is depicted with certain ``clumps'' (there are 5 in the bottom row, 5 in the middle row, and 3 in the top row).  Why are these clumped as they are?  What distinguishes each clump?  

\item 
Broadly speaking, how was the ordering (left to right) of faces chosen in each row of the face lattice?  

\end{itemize}

% \begin{center}
% \includegraphics[width=2in]{week8-a3-polytope.pdf}
% \end{center}

\begin{proof}

\end{proof}


\item 
Prove that any subset of the vertices of the $d$-simplex
$$S_d = \Conv\{0, e_1, \ldots, e_d\}$$
is the $V$-description of a face of $S_d$ (here, $e_i$ denotes the $i$-th standard basis vector).  

\begin{proof}

\end{proof}


\item 
Find a formula for the number of ridges of the $d$-dimensional cube $C_d$.  

\begin{proof}

\end{proof}


\item 
Complete and write up the last part of Problem~(D2).  

\begin{proof}

\end{proof}

\end{enumerate}



\bigskip
\noindent
\textbf{Challenge problems.}
Challenge problems are not required for submission, but bonus points will be awarded for submitting a partial attempt or a complete solution.  

\begin{enumerate}[(C1)]
\item 
Locate a halfspace defining each face of the permutohedron $P_n$.  

\begin{proof}

\end{proof}

\end{enumerate}

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\end{document}

