Cycle index monomial of a permutation:

Cycle index polynomial for the alternating group on five points:

Cycle index monomial of permutations:

Specify the size of the domain of action:

Cycle index polynomial of permutation groups:

Specify the size of the domain of action:

Cycle index polynomials are essential in Pólya's theory of counting. The classical example is counting how many necklaces can be formed with beads of different colors:

Suppose there is a necklace with 10 beads, invariant under cyclic rotations:

Suppose that there are beads of three colors, denoted by r, g, b:

For instance, there are 252 necklaces with three r beads, five g beads and two b beads:

This can be checked by actual construction of the necklaces:

If the necklace is considered to also be invariant under reflections along a diameter, then the symmetry group is dihedral:

Now the counts are different:

The identity permutation has degree zero and hence does not move any point:

If it acts on a set with four points, then the cycle index polynomial reflects the existence of four singletons:

Each point not moved contributes a multiplication by the first variable:

Missing variables are effectively replaced by 1:

The cycle index polynomial of a direct product of groups is the product of the cycle index polynomial of the groups: