Necklace (combinatorics)

In combinatorics, a k-ary necklace of length n is an equivalence class (a grouping for which there exists an equivalence relation) of n-character strings over an alphabet of size k, taking all rotations as equivalent. It represents a structure with n circularly connected beads which have k available colors.

Possible patterns of bracelets of length n
corresponding to the k-th integer partition
(set partitions up to rotation and reflection)

The 3 bracelets with 3 red and 3 green beads. The one in the middle is chiral, so there are 4 necklaces.
Compare box(6,9) in the triangle.

The 11 bracelets with 2 red, 2 yellow and 2 green beads. The leftmost one and the four rightmost ones are chiral, so there are 16 necklaces.
Compare box(6,7) in the triangle.

16 Tantrix tiles, corresponding to the 16 necklaces with 2 red, 2 yellow and 2 green beads.

A k-ary bracelet, also referred to as a turnover (or free) necklace, is a necklace such that strings may also be equivalent under reflection. That is, given two strings, if each is the reverse of the other, they belong to the same equivalence class. For this reason, a necklace might also be called a fixed necklace to distinguish it from a turnover necklace.

Formally, one may represent a necklace as an orbit of the cyclic group acting on n-character strings, and a bracelet as an orbit of the dihedral group. One can count these orbits, and thus necklaces and bracelets, using Pólya's enumeration theorem.

Equivalence classes

Number of necklaces

There are

${\displaystyle N_{k}(n)={\frac {1}{n}}\sum _{d\mid n}\varphi (d)k^{\frac {n}{d}}}$

different k-ary necklaces of length n, where ${\displaystyle \varphi }$ is Euler's totient function.[1] This follows directly from Pólya's enumeration theorem applied to the action of the cyclic group ${\displaystyle C_{n}}$ acting on the set of all functions ${\displaystyle f:\{1,\ldots ,n\}\to \{1,\ldots ,k\}}$. There are also

${\displaystyle L_{k}(n)={\frac {k!}{n}}\sum _{d\mid n}\varphi (d)S({\frac {n}{d}},k)}$

different necklaces of length n with exactly k different colored beads, where ${\displaystyle S(n,k)}$ are the Stirling number of the second kind.

${\displaystyle N_{k}(n)}$ (sequence A054631 in the OEIS) and ${\displaystyle L_{k}(n)}$ (sequence A087854 in the OEIS) are related via the Binomial coefficients:

${\displaystyle N_{k}(n)=\sum _{j=1}^{k}{\binom {k}{j}}L_{k}(n)}$

and

${\displaystyle L_{k}(n)=\sum _{j=1}^{k}(-1)^{k-j}{\binom {k}{j}}N_{k}(n)}$

Number of bracelets

There are total

${\displaystyle B_{k}(n)={\begin{cases}{\tfrac {1}{2}}N_{k}(n)+{\tfrac {1}{4}}(k+1)k^{\frac {n}{2}}&{\text{if }}n{\text{ is even}}\\[10px]{\tfrac {1}{2}}N_{k}(n)+{\tfrac {1}{2}}k^{\frac {n+1}{2}}&{\text{if }}n{\text{ is odd}}\end{cases}}}$

different k-ary bracelets of length n, where N_k(n) is the number of k-ary necklaces of length n. This follows from Pólya's method applied to the action of the dihedral group ${\displaystyle D_{n}}$.

Case of distinct beads

For a given set of n beads, all distinct, the number of distinct necklaces made from these beads, counting rotated necklaces as the same, is n!/n = (n − 1)!. This is because the beads can be linearly ordered in n! ways, and the n circular shifts of such an ordering all give the same necklace. Similarly, the number of distinct bracelets, counting rotated and reflected bracelets as the same, is n!/2n, for n ≥ 3.

If the beads are not all distinct, having repeated colors, then there are fewer necklaces (and bracelets). The above necklace-counting polynomials give the number necklaces made from all possible multisets of beads. Polya's pattern inventory polynomial refines the counting polynomial, using variable for each bead color, so that the coefficient of each monomial counts the number of necklaces on a given multiset of beads.

Aperiodic necklaces

An aperiodic necklace of length n is a rotation equivalence class having size n, i.e., no two distinct rotations of a necklace from such class are equal.

According to Moreau's necklace-counting function, there are

${\displaystyle M_{k}(n)={\frac {1}{n}}\sum _{d\mid n}\mu (d)k^{\frac {n}{d}}}$

different k-ary aperiodic necklaces of length n, where μ is the Möbius function. The two necklace-counting functions are related by: ${\displaystyle N_{k}(n)\ =\ \sum \nolimits _{d|n}M_{k}(d),}$ where the sum is over all divisors of n, which is equivalent by Möbius inversion to ${\displaystyle M_{k}(n)\ =\ \sum \nolimits _{d|n}N_{k}(d)\,\mu ({\tfrac {n}{d}}).}$

Each aperiodic necklace contains a single Lyndon word so that Lyndon words form representatives of aperiodic necklaces.

See also

• Lyndon word
• Inversion (discrete mathematics)
• Necklace problem
• Necklace splitting problem
• Permutation
• Proofs of Fermat's little theorem#Proof by counting necklaces
• Forte number, a representation of binary bracelets of length 12 used in atonal music.

References

1. Weisstein, Eric W. "Necklace". MathWorld.

External links

• Weisstein, Eric W. "Necklace". MathWorld.
• Info on necklaces, Lyndon words, De Bruijn sequences