Bracelet (combinatorics)

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 of up to k different 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 then 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.

Technically, one may classify a necklace as an orbit of the action of the cyclic group on n-character strings, and a bracelet as an orbit of the dihedral group's action. This enables application of Pólya enumeration theorem for enumeration of necklaces and bracelets.

There are

different k-ary necklaces of length n, where φ is Euler's totient function.

There are

different k-ary bracelets of length n, where N_k(n) is the number of k-ary necklaces of length n.

If there are n beads, all distinct, on a necklace joined at the ends, then the number of distinct orderings on the necklace, after allowing for rotations, is n!/n, for n > 0. This may also be expressed as (n − 1)!. This number is less than the general case, which lacks the requirement that each bead must be distinct.

An intuitive justification for this can be given. If there is a line of n distinct objects ("beads"), the number of combinations would be n!. If the ends are joined together, the number of combinations are divided by n, as it is possible to rotate the string of n beads into n positions.

...
Wikipedia