In
combinatorics, a k-ary necklace of length n is an
equivalence class 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.
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 groupacting on n-character strings over an alphabet of size k, 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.
different k-ary necklaces of length n, where is
Euler's totient function.[1]
When the beads are restricted to particular color
multiset, where is the number of beads of color , there are
different necklaces made of all the beads of . [2]
Here and is the
multinomial coefficient.
These two formulas follow directly from
Pólya's enumeration theorem applied to the action of the cyclic group acting on the set of all functions .
If all k colors must be used, the count is
The number of different k-ary bracelets of length n (sequence A081720 in the
OEIS) is
where Nk(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.
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.
different k-ary aperiodic necklaces of length n, where μ is the
Möbius function. The two necklace-counting functions are related by: where the sum is over all divisors of n, which is equivalent by
Möbius inversion to
Each aperiodic necklace contains a single
Lyndon word so that Lyndon words form
representatives of aperiodic necklaces.