组合数学 (Fall 2011)/Pólya's theory of counting

Groups

A group [math]\displaystyle{ (G,\cdot) }[/math] is set [math]\displaystyle{ G }[/math] along with a binary operator [math]\displaystyle{ \cdot }[/math] which satisfies the following axioms:

• closure: [math]\displaystyle{ \forall g,h\in G, g\cdot h \in G }[/math];
• associativity: [math]\displaystyle{ \forall f,g,h\in G, f\cdot(g\cdot h)=(f\cdot g)\cdot h }[/math];
• identity: there exists a special element [math]\displaystyle{ e\in G }[/math], called the identity, such that [math]\displaystyle{ e\cdot g=g }[/math] for any [math]\displaystyle{ g\in G }[/math];
• inverse: [math]\displaystyle{ \forall g\in G }[/math], there exists an [math]\displaystyle{ h\in G }[/math] such that [math]\displaystyle{ g\cdot h=e }[/math], and we denote that [math]\displaystyle{ h=g^{-1} }[/math].

Group action

Definition (group action)

A group action of a group [math]\displaystyle{ G }[/math] on a set [math]\displaystyle{ X }[/math] is a binary operator: [math]\displaystyle{ \circ:G\times X\rightarrow X }[/math] satisfying: Associativity: [math]\displaystyle{ (g\cdot h)\circ x=g\circ (h\circ x) }[/math] for all [math]\displaystyle{ g,h\in G }[/math] and [math]\displaystyle{ x\in X }[/math]; Identity: [math]\displaystyle{ e\circ x=x }[/math] for all [math]\displaystyle{ x\in X }[/math].

Burnside's Lemma

Counting orbits

Burnside's Lemma

Let [math]\displaystyle{ G }[/math] be a permutation group acting on a set [math]\displaystyle{ X }[/math]. For each [math]\displaystyle{ \pi\in G }[/math], let [math]\displaystyle{ X_\pi=\{x\in X\mid \pi\circ x=x\} }[/math] be the set of elements invariant under action by [math]\displaystyle{ \pi }[/math]. The number of orbits, denoted [math]\displaystyle{ |X/G| }[/math], is [math]\displaystyle{ |X/G|=\frac{1}{|G|}\sum_{\pi\in G}|X_{\pi}|. }[/math]

Proof. Let [math]\displaystyle{ A(\pi,x)=\begin{cases}1 & \pi\circ x=x,\\ 0 & \text{otherwise.}\end{cases} }[/math] Basically [math]\displaystyle{ A(\pi,x) }[/math] indicates whether [math]\displaystyle{ x }[/math] is invariant under action of [math]\displaystyle{ \pi }[/math]. We can think of [math]\displaystyle{ A }[/math] as a matrix indexed by [math]\displaystyle{ G\times X }[/math]. An important observation is that the row sums and column sums of [math]\displaystyle{ A }[/math] are [math]\displaystyle{ |X_\pi| }[/math] and [math]\displaystyle{ |G_x| }[/math] respectively, namely, [math]\displaystyle{ |X_\pi|=\sum_{x\in X}A(\pi,x) }[/math] and [math]\displaystyle{ |G_x|=\sum_{\pi\in G}A(\pi,x) }[/math]. Then a double counting gives that [math]\displaystyle{ \sum_{\pi\in G}|X_\pi|=\sum_{\pi\in G}\sum_{x\in X}A(\pi,x)=\sum_{x\in X}\sum_{\pi\in G}A(\pi,x)=\sum_{x\in X}|G_x| }[/math]. [math]\displaystyle{ \square }[/math]