Pólya Enumeration Theorem

Cycle index of symmetry group → generating polynomial for distinct colorings

Shape: Colors:

Z(G; a_1,...,a_k) = (1/|G|) Σ_{g∈G} a_1^{c_1(g)} a_2^{c_2(g)} ... a_n^{c_n(g)}

Substituting a_i = k (k colors): gives distinct colorings count

The cycle index records the cycle structure of every group element. Substituting k for each variable gives the total number of distinct colorings with k colors. Full polynomial tracks color distribution.