Let j_k(α) denote the number of cycles of length k for a permutation α expressed as a product of disjoint cycles. The cycle index Z(X) of a permutation group X of order m = left bracketing bar X right bracketing bar and degree d is then the polynomial in d variables x_1, x_2, ..., x_d given by the formula Z(X) = 1/( left bracketing bar X right bracketing bar ) sum_(α element X) product_(k = 1)^d x_k^(j_k(α)). The cycle index of a permutation group X is implemented as CycleIndexPolynomial[perm, {x1, ..., xn}], which returns a polynomial in x_i.

alternating group | cycle graph | cycle polynomial | cyclic group | dihedral group | permutation cycle | permutation group | Pólya enumeration theorem | simple graph | symmetric group | symmetric polynomial