Necklace polynomial

In combinatorial mathematics, the necklace polynomial, or Moreau's necklace-counting function, introduced by C. Moreau (1872), counts the number of distinct necklaces of n colored beads chosen out of α available colors, arranged in a cycle. Unlike the usual problem of graph coloring, the necklaces are assumed to be aperiodic (not consisting of repeated subsequences), and counted up to rotation (rotating the beads around the necklace counts as the same necklace), but without flipping over (reversing the order of the beads counts as a different necklace). This counting function also describes the dimensions in a free Lie algebra and the number of irreducible polynomials over a finite field.