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COUNTING SYMMETRIC BRACELETS
Published online by Cambridge University Press: 22 August 2013
Abstract
An $r$-ary necklace (bracelet) of length
$n$ is an equivalence class of
$r$-colourings of vertices of a regular
$n$-gon, taking all rotations (rotations and reflections) as equivalent. A necklace (bracelet) is symmetric if a corresponding colouring is invariant under some reflection. We show that the number of symmetric
$r$-ary necklaces (bracelets) of length
$n$ is
$\frac{1}{2} (r+ 1){r}^{n/ 2} $ if
$n$ is even, and
${r}^{(n+ 1)/ 2} $ if
$n$ is odd.
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- Research Article
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- Copyright
- Copyright ©2013 Australian Mathematical Publishing Association Inc.
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