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. The necklaces are assumed to be aperiodic (not consisting of repeated subsequences), and the counting is done "without flipping over" (without reversing the order of the beads). This counting function describes, among other things, the number of free Lie algebras and the number of irreducible polynomials over a finite field.

Definition[]

The necklace polynomials are a family of polynomials ${\displaystyle M(\alpha ,n)}$ in the variable ${\displaystyle \alpha }$ such that

${\displaystyle \alpha ^{n}\ =\ \sum _{d\,|\,n}d\,M(\alpha ,d).}$

By Möbius inversion they are given by

${\displaystyle M(\alpha ,n)\ =\ {1 \over n}\sum _{d\,|\,n}\mu \!\left({n \over d}\right)\alpha ^{d},}$

where ${\displaystyle \mu }$ is the classic Möbius function.

A closely related family, called the general necklace polynomial or general necklace-counting function, is:

${\displaystyle N(\alpha ,n)\ =\ \sum _{d|n}M(\alpha ,d)\ =\ {\frac {1}{n}}\sum _{d\,|\,n}\phi \!\left({n \over d}\right)\alpha ^{d},}$

where ${\displaystyle \phi }$ is Euler's totient function.

Applications[]

The necklace polynomials ${\displaystyle M(\alpha ,n)}$ appear as:

• The number of aperiodic necklaces (or equivalently Lyndon words) which can be made by arranging n colored beads having α available colors. Two such necklaces are considered equal if they are related by a rotation (but not a reflection). Aperiodic refers to necklaces without rotational symmetry, having n distinct rotations. The polynomials ${\displaystyle N(\alpha ,n)}$ give the number of necklaces including the periodic ones: this is easily computed using Pólya theory.
• The dimension of the degree n piece of the free Lie algebra on α generators ("Witt's formula"^[1]). Here ${\displaystyle N(\alpha ,n)}$ should be the dimension of the degree n piece of the corresponding free Jordan algebra.
• The number of distinct words of length n in a Hall set. Note that the Hall set provides an explicit basis for a free Lie algebra; thus, this is the generalized setting for the above.
• The number of monic irreducible polynomials of degree n over a finite field with α elements (when ${\displaystyle \alpha =p^{d}}$ is a prime power). Here ${\displaystyle N(\alpha ,n)}$ is the number of polynomials which are primary (a power of an irreducible).
• The exponent in the cyclotomic identity.

Despite the fact that the polynomials appear in these various settings, the precise relationships between these remain mysterious or unknown. For example, there is no known bijection between the irreducible polynomials and the Lyndon words.^[2]

Relations between M and N[]

The polynomials for M and N are easily related in terms of Dirichlet convolution of arithmetic functions ${\displaystyle f(n)*g(n)}$, regarding ${\displaystyle \alpha }$ as a constant.

• The formula for M gives ${\displaystyle n\,M(n)\,=\,\mu (n)*\alpha ^{n}}$,
• The formula for N gives ${\displaystyle n\,N(n)\,=\,\phi (n)*\alpha ^{n}\,=\,n*\mu (n)*\alpha ^{n}}$.
• Their relation gives ${\displaystyle N(n)\,=\,1*M(n)}$ or equivalently ${\displaystyle n\,N(n)\,=\,n*(n\,M(n))}$, since n is completely multiplicative.

Any two of these imply the third, for example:

${\displaystyle n*\mu (n)*\alpha ^{n}\,=\,n\,N(n)\,=\,n*(n\,M(n))\quad \Longrightarrow \quad \mu (n)*\alpha ^{n}=n\,M(n)}$

by cancellation in the Dirichlet algebra.

Examples[]

${\displaystyle {\begin{aligned}M(1,n)&=0{\text{ if }}n>1\\[6pt]M(\alpha ,1)&=\alpha \\[6pt]M(\alpha ,2)&={\tfrac {1}{2}}(\alpha ^{2}-\alpha )\\[6pt]M(\alpha ,3)&={\tfrac {1}{3}}(\alpha ^{3}-\alpha )\\[6pt]M(\alpha ,4)&={\tfrac {1}{4}}(\alpha ^{4}-\alpha ^{2})\\[6pt]M(\alpha ,5)&={\tfrac {1}{5}}(\alpha ^{5}-\alpha )\\[6pt]M(\alpha ,6)&={\tfrac {1}{6}}(\alpha ^{6}-\alpha ^{3}-\alpha ^{2}+\alpha )\\[6pt]M(\alpha ,p)&={\tfrac {1}{p}}(\alpha ^{p}-\alpha )&{\text{ if }}p{\text{ is prime}}\\[6pt]M(\alpha ,p^{N})&={\tfrac {1}{p^{N}}}(\alpha ^{p^{N}}-\alpha ^{p^{N-1}})&{\text{ if }}p{\text{ is prime}}\end{aligned}}}$

For ${\displaystyle \alpha =2}$, starting with length zero, these form the integer sequence

1, 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, ... (sequence A001037 in the OEIS)

Identities[]

The polynomials obey various combinatorial identities, given by Metropolis & Rota:

${\displaystyle M(\alpha \beta ,n)=\sum _{\operatorname {lcm} (i,j)=n}\gcd(i,j)M(\alpha ,i)M(\beta ,j),}$

where "gcd" is greatest common divisor and "lcm" is least common multiple. More generally,

${\displaystyle M(\alpha \beta \cdots \gamma ,n)=\sum _{\operatorname {lcm} (i,j,\ldots ,k)=n}\gcd(i,j,\cdots ,k)M(\alpha ,i)M(\beta ,j)\cdots M(\gamma ,k),}$

which also implies:

${\displaystyle M(\beta ^{m},n)=\sum _{\operatorname {lcm} (j,m)=nm}{\frac {j}{n}}M(\beta ,j).}$

Cyclotomic identity[]

${\displaystyle {1 \over 1-\alpha z}\ =\ \prod _{j=1}^{\infty }\left({1 \over 1-z^{j}}\right)^{M(\alpha ,j)}}$

References[]

• Moreau, C. (1872), "Sur les permutations circulaires distinctes (On distinct circular permutations)", Nouvelles Annales de Mathématiques, Série 2 (in French), 11: 309–31, JFM 04.0086.01

• Metropolis, N.; Rota, Gian-Carlo (1983), "Witt vectors and the algebra of necklaces", Advances in Mathematics, 50 (2): 95–125, doi:10.1016/0001-8708(83)90035-X, ISSN 0001-8708, MR 0723197, Zbl 0545.05009

• Reutenauer, Christophe (1988). "Mots circulaires et polynomies irreductibles". Ann. Sc. Math. Quebec. 12 (2): 275–285.