Moreau's necklace-counting function

In combinatorial mathematics, the necklace polynomials, or (Moreau's) necklace-counting function are the polynomials M(α,n) in α such that

By Möbius inversion they are given by

where μ is the classic Möbius function.

The necklace polynomials are closely related to the functions studied by C. Moreau (1872), though they are not quite the same: Moreau counted the number of necklaces, while necklace polynomials count the number of aperiodic necklaces.

The necklace polynomials appear as:

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