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\begin{document}
\flushright{\noindent 1-4: Blake Kruithof
\\5-6: Joseph Crews
\\7-9: Devin Dixon}
\center \section{\LaTeX\ Challenge}
Here are some symbols:
$$\rho, \vee, \mathcal{P}, \rightarrow, \varepsilon, \Phi, \beta, \leq, \geq, \sum, \textsf{smg}$$
\begin{equation}
\sum_{i=1}^{4} i^2=\text{Some number}
\end{equation}
\begin{align*}
\sum_{i=1}^{4} i^2 &=\text{Some number}\\
&= x+y\\
&= y+x
\end{align*}
\begin{theorem}
This is a sample theorem.
\end{theorem}
\begin{lemma}
This is a sample lemma. Also $N+1=M$.
\begin{enumerate}
\item Item 1.
\item Item 2.
\end{enumerate}
\end{lemma}
\begin{definition}
This is a sample definition. Also $N+1=M$.
\end{definition}
\begin{example}
This is a sample definition.
\begin{itemize}
\item Item 1.
\item Item 2.
\end{itemize}
\end{example}
\vspace{124pc}
--------------------------------------------------------------------------
\noindent All of this is to say that each legal shape is the ``partition diagram" (or ``Ferrer's diagram") for a partition with no more than $k$ parts, and the largest part of size at most $N-k$. We call this partition, or its corresponding shape, a $k\times(N-k)$ {\it partition}.
A {\emph{domino move}} is the addition or removal of
\begin{itemize}
\item A domino pair, or
\item A singleton in the upper right corner only.
\end{itemize}
\vspace{7pc}
\noindent {\textbf{Results.}} We now connect the ideas of mountain- and valley-ization of paths with the classical concepts of {\emph {distributive}} and {\emph {modular lattices}}. (For reference, see [4].) In our setting, if $\rho$ denotes a rank function for some poset $R$, then for any path $\mathcal{P}$ from \textbf{s} to \textbf{t}, we have
$$\rho(\textbf{t})-\rho(\textbf{t})=\sum_{i\in I} ({a_{i}}(\mathcal{P})-d_{i}(\mathcal{P})),$$
\vspace{7pc}
\begin{theorem} Let \textbf s and \textbf t be elements of a topographically balanced lattice L and let $\mathcal{P}$ be any simple path in L from \textbf s to \textbf t. Then the following are equivalent:
\begin{enumerate}
\item ${\mathcal{P}}$ is a path of shortest length from \textbf s to \textbf t.
\item $\ell (\mathcal{P})=2\rho ($\textbf s$ \vee $\textbf t$)-\rho($\textbf s$)-\rho($\textbf t$)=\rho($\textbf s$)+\rho($\textbf t$)-2\rho($ \textbf s $ \wedge $ \textbf t $) $.
\item $\mathcal{P}_{mountain}$ has apex $\textbf u= \textbf s \vee \textbf t$ and $\mathcal{P}_{valley}$ has nadir $\textbf v= \textbf s \wedge \textbf t$.
\end{enumerate}
\end{theorem}
\vspace{7pc}
\begin{example}
Let $\sigma=(3,3)\in L_A(2,4)$. As in Example 5.5, we see that the diagonalization of $\sigma$ is $\textbf s=(0,1,2,2,1)$, and, according to the same picture, we see that
$$(0,1,2,2,1)= \epsilon_2+ 2\epsilon_3+ 2s\epsilon_4+\epsilon_5,$$ \flushright{(5.5)}
\end{example}
\vspace{7pc}
\begin{example} As in Example 5.12, but now considering coordinates, we have that (0,1,2,2,1) $\xrightarrow{\Phi}$ (0,1,1,2,1). And, equivalently Equation 5.5, one can confirm with \textit{D}(2,4) in Example 5.5 that
\begin{align*}
(0,1,1,2,1) &=(0,-1,-1,0,0)+2(-1,0,0,0,0)+2(1,1,0,0,0)+(0,0,1,1,0) \\\indent+[(0,0,1,1,1)]\\ &=\beta_{2} + 2\beta_{3} + 2\beta_{4} + \beta_{5} + \textsf{m}
\end{align*}
\end{example}
\vspace{9pc}
\begin{itemize}
\item \emph{For any join irreducible} $\textbf{ x }$ $\emph{L, the set } $\{\textbf{y}$ \in \emph{ K } \mid \emph{ x } \leq_{\emph{L}} $\textbf{y}\}$ \emph{ has a unique} \\\emph{minimal element} {$\textbf{ w_{x}}$}
\item \emph{the function} \phi : \emph{Q} \rightarrow \emph{P given by } \phi(\textbf{x}) := {$\textbf{ w_{x}}$} \emph{is a vertex-color-preserving} \\\emph{bijection, and if} {$\textbf{u}$} \leq_{\emph{Q}} {$\textbf{v}$} \emph{ then } \phi(\textbf{u}) \leq_{\emph{P}} \phi(\textbf{v}) \emph{. Now let Q' be the set P and} \\\emph{declare that } \phi(\textbf{u}) \leq_{\emph{Q'}} \phi(\textbf{v}) \emph{ if and only if } {$\textbf{u }$} \leq_{\emph{Q}} {$\textbf{v}$} \emph{. Then Q' is a weak} \\\emph{subset of P and Q' } $\cong$ \emph{ as vertex-colored posets.}
\end{itemize}
\vspace{9pc}
\noindent\textbf{Example 5.14.} We now connect examples 5.12 and 5.13. Considering $D(2,4)$, let \newline $\textsf{s}= (0,1,1,2,1) \in D_\textsf{A}^{\textsf {diag}}$ as in Equation 5.6, with $\textsf{m}=(0,0,1,1,1)$. Then
\begin{align*}
P^{-1}(\textsf{s}-\textsf{m})=
\begin{pmatrix}
0 & 0 & -1 & 1 & 0\\
0 & -1 & 0 & 1 & 0\\
0 & -1 & 0 & 0 & 1\\
-1 & 0 & 0 & 0 & 1\\
-1 & 0 & 0 & 0 & 0\\
\end{pmatrix}^{-1}
\begin{pmatrix}
0\\
1\\
0\\
1\\
0\\
\end{pmatrix}
=
\begin{pmatrix}
0\\
1\\
2\\
2\\
1\\
\end{pmatrix}
\end{align*}
So, as in Equation 5.6, we see that $\textsf{s}=\beta_2+2\beta_3+2\beta_4+\beta_5$.
\vspace{9pc}
\noindent\textbf{A Domino solution to (1)-(3).} Given partitions $\sigma$ and $\tau$ in $D_\textsf{A}$ with corresponding diagonal coordinates \textsf{s} and \textsf{t}, and with \textsf{m} be the minimum element in $D_\textsf{A}$, consider the expansions $(\textsf{s}$-$\textsf{m})=\sum c_i\beta_i$ and $(\textsf{t}$-$\textsf{m}) =\sum d_i\beta_i$ which can be found by $P^{-1}$(\textsf{s}$-$\textsf{m}) and $P^{-1}$(\textsf{t}$-$\textsf{m}). We use the mountain path defined in Section 2 to determine the length of the shortest path from $\sigma$ to $\tau$ and we prescribe that path. Let
\begin{align*}
\textsf {S}&=\{i^{c_i} | 1 \leq i \leq N-1\}, \text{ and}\\
\textsf{T}&=\{i^{d_i} | 1 \leq i \leq N-1\}
\end{align*}
\vspace{9pc}
\noindent\textbf{Example 5.16.} Let $\sigma$ and $\tau$ be as in Example 5.15. Then the path from $\sigma$ to $\sigma \vee \tau$ consists of the edge 2. The path from $\tau$ to $\sigma \vee \tau$ consists of traversing edge 5 then edge 4. Therefore the path beginning at $\sigma$ consisting of edges 2, 4, and then 5 in gives a path from $\sigma$ to $\tau$. That is,
\begin{enumerate}
\item[0.] $\sigma = (4,4) \longleftrightarrow \textsf{s}=(1,2,2,2,1)$
\item[1.] $\textsf{s}^1=\textsf{s}+\beta_2= (1,1,1,2,1)=\textsf{t}^2\longleftrightarrow\sigma\vee\tau$
\item[2.] $\textsf{t}^1=\textsf{t}^2-\beta_4=(0,0,1,2,1)$
\item[3.] $\textsf{t}=\textsf{t}^1-\beta_5=(0,0,0,1,1)\longleftrightarrow\tau=(1,1)$
\end{enumerate}
\end{document}
