Necklace polynomial

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

\alpha^n = \sum_{d\,|\,n} d \, M(\alpha, d).

By Möbius inversion they are given by

 M(\alpha,n) = {1\over n}\sum_{d\,|\,n}\mu\left({n \over d}\right)\alpha^d

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"[1]);
  • 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.[1]


  • M(α,1) = α
  • M(α,2) = (α2 − α)/2
  • M(α,3) = (α3 − α)/3
  • M(α,4) = (α4 − α2)/4
  • M(α,5) = (α5 − α)/5
  • M(α,6) = (α6 − α3 − α2 + α)/6
  • M(α,pn) = (αpn − αpn − 1)/pn for p prime
  • \displaystyle M(\alpha\beta, n)=\sum_{[i,j]=n}(i,j)M(\alpha,i)M(\beta,j) where (i,j) is the highest common factor of i and j and [i,j] is their least common multiple.

See also[edit]


  1. ^ a b Lothaire, M. (1997). Combinatorics on words. Encyclopedia of Mathematics and Its Applications 17. Perrin, D.; Reutenauer, C.; Berstel, J.; Pin, J. E.; Pirillo, G.; Foata, D.; Sakarovitch, J.; Simon, I.; Schützenberger, M. P.; Choffrut, C.; Cori, R.; Lyndon, Roger; Rota, Gian-Carlo. Foreword by Roger Lyndon (2nd ed.). Cambridge University Press. pp. 79,84. ISBN 0-521-59924-5. MR 1475463. Zbl 0874.20040.