Burnside lemma / Pólya Theorem

• \ (A \) and \ (B \) is a finite set

• \ (X = B ^ A \ ) represents all \ (A \) to \ (B \) mapping

• \ (G \) is \ (A \) permutation group on, \ (X-/ G \) represents \ (G \) acting \ (X-\) set of equivalence classes on the

• \(X^g=\{x|x\in X,g(x)=x\}\)

Burnside lemma

\[ |X/G|=\frac{1}{|G|}\sum_{g\in G}|X^g| \]

• \ (c (g) \) represents substitution \ (G \) quantities can be split into disjoint cyclic permutation of

Pólya Theorem

\[ |X/G|=\frac{1}{|G|}\sum_{g\in G}|B|^{c(g)} \]