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\title{Final exam (take-home)}
\author{Algebraic Combinatorics (Math 68)}
\date{\vspace{-.5em}Due November 27, 2019}

\begin{document}
    \maketitle
    As this is an exam, \textbf{you are not allowed to give or receive any help}, except from the instructor.
    However, \textbf{you are allowed to use the lectures notes} and any assignment you have completed for this course. Other references are not allowed.
    
    You must write the appropriate justification as part of the solutions.\\
    
    Please, \textbf{turn in your solutions by email} (since we won't meet over the period of time for the exam).\\ 
    
   	\begin{enumerate}
   		\item (25 points)\quad Prove this interpretation of Erd\H{o}s-Szekeres theorem: Any permutation of length greater than $n^2$ must contain either an increasing sequence of length at least $n$ or a decreasing sequence of length at least $n$. Prove that this bound is sharp (i.e. that there is a permutation of length $n^2$ with no increasing nor decreasing sequence of length $n+1$).\\
   		
   		\item (25 points)\quad Let $A_n$ denote the alternating subgroup of $S_n$ (i.e. the group of even permutations). Let $\sigma \in S_n$ have cycle type $(\lambda_1,\ldots, \lambda_l)$.
   		\begin{enumerate}
   			\item Show that $\sigma \in A_n$ if and only if $n-l$ is even.
   			\item Explain why $A_4$ has four irreducible representations.
   			\item Do all characters of $A_n$ have integer values? Why?
   			\item Give two important differences between the table of characters of $A_n$ and of $S_n$.\\
   		\end{enumerate}
   	
	   	\item (20 points)\quad True or False: If every chain and every antichain of a poset $P$ is finite, then $P$ is finite (as a set). You must justify your answer.\\
	   	
	   	\item (10 points)\quad How many compositions of 17 use only parts of length $2$ and $3$?\\
	   	
	   	\item (20 points)\quad How many distinct regular tetrahedra are there under rotation if the faces are colored from a set with $r$ colors. Also, give a numerical answer for $r = 1,2,3,4,5$.\\
	   	
	\end{enumerate}

    \begin{center}
    	\textbf{Good luck!}
    \end{center}
\end{document}