Necklace Polynomial

Necklace Polynomial

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:

• the number of aperiodic necklaces (also called Lyndon words) that can be made by arranging n beads the color of each of which is chosen from a list of α colors (One respect in which the word "necklace" may be misleading is that if one picks such a necklace up off the table and turns it over, thus reversing the roles of clockwise and counterclockwise, one gets a different necklace, counted separately, unless the necklace is symmetric under such reflections.);
• the dimension of the degree n piece of the free Lie algebra on α generators ("Witt's formula");
• the number of monic irreducible polynomials of degree n over a finite field with α elements (when α is a prime power);
• the exponent in the cyclotomic identity;
• The number of Lyndon words of length n in an alphabet of size α.