Necklace polynomial

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:

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

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

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.

Although these various types of objects are all counted by the same polynomial, their precise relationships remain mysterious or unknown. For example, there is no canonical bijection between the irreducible polynomials and the Lyndon words.[2] However, there is a non-canonical bijection that can be constructed as follows. For any degree n monic irreducible polynomial over a field F with α elements, its roots lie in a Galois extension field L with ${\displaystyle \alpha ^{n}}$ elements. One may choose an element ${\displaystyle x\in L}$ such that ${\displaystyle \{x,\sigma x,...,\sigma ^{n-1}x\}}$ is an F-basis for L (a normal basis), where σ is the Frobenius automorphism ${\displaystyle \sigma y=y^{\alpha }}$. Then the bijection can be defined by taking a necklace, viewed as an equivalence class of functions ${\displaystyle f:\{1,...,n\}\rightarrow F}$, to the irreducible polynomial ${\displaystyle \phi (T)=(T-y)(T-\sigma y)\cdots (T-\sigma ^{n-1}y)\in F[T]}$, where ${\displaystyle y=f(1)x+f(2)\sigma x+\cdots +f(n)\sigma ^{n-1}x}$. Different cyclic rearrangements of f, i.e. different representatives of the same equivalence class, yield cyclic rearrangements of the factors of ${\displaystyle \phi (T)}$, so this correspondence is well-defined.[3]

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.

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

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)

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:

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

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

• 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.