\chapter{Algebraic combinatorics}

This chapter contains an assortment of combinatorial results for algebraic structures and introduces structures made to assist in combinatorics. For various structures in abstract algebra, there exist intriguing results relating to cardinalities of their sets, particular subsets, or other sets they act on. In the spirit of combinatorics, we seek to translate combinatorial problems into algebraic problems to invoke these methods.

\section{Burnside's lemma}

Group actions are very powerful for creating combinatorial arguments, among the most prominent being \emph{Burnside's lemma}.

We have established that orbits are either equal or disjoint, and elements of orbits  of $x$ have a bijection to cosets of the stabilizer of $x$. means that the sum of these stabilizer coset cardinalities is the sum of the orbit cardinalitites.

\begin{corollary}
Let $G$ be a finite group acting on $S$, then the following holds where $B$ is a set constructed by choosing one element from each orbit with over one element.
\[ |S| = |S^G| + \sum_{x \in B} |G / G_x| \]
\end{corollary}


Using this corollary, one can derive \emph{Burnside's lemma}. It states that we can count the amount of unique orbits by finding the 'average' of each '$g$-invariant set of elements' (set elements where the group action with $g$ has no effect). It is an indispensable tool in combinatorics; particularly for counting how many symmetrically unique colorings there are on an $n$-gon.

\begin{lemma}[Burnside's lemma]
	\[ |S / G|  = \frac{\sum_{g \in G}|S^g|}{|G|}\]
	\[ S^g = \{s \in S : gs=s\}\]
\end{lemma}


\section{Pólya enumeration theorem}

\section{Matroids}


\section{Young Tableaux}