Moreau's necklace-counting function
In
combinatorial mathematics,
Moreau's necklace-counting function
where μ is the classic Möbius function, counts the number of "necklaces" asymmetric under rotations 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.
This function is involved in the cyclotomic identity.
References
- C. Moreau. Sur les permutations circulaires distincts. Nouv. Ann. Math., volume 11, pages 309-314, 1872.
- Nicholas Metropolis & Gian-Carlo Rota. Witt Vectors and the Algebra of Necklaces. Advances in Mathematics, volume 50, number 2, pages 95-125, 1983.