Bracelet (combinatorics)

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In combinatorics, a k-ary bracelet of length n is the equivalence class of all n-character strings over an alphabet of size k, taking reverse and all rotations as equivalent. A bracelet, also referred to as a turnover necklace, represents a structure with n circulary connected beads of k different colors, which (unlike a necklace) can be turned over.

There are

$B_k(n) = \begin{cases} {1\over 2}N_k(n) + {1\over 4}(k+1)k^{n/2} & \mbox{if }n\mbox{ is even} \\ \\ {1\over 2}N_k(n) + {1 \over 2}k^{(n+1)/2} & \mbox{if }n\mbox{ is odd} \end{cases}$