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COUNTING SYMMETRIC BRACELETS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 89 / Issue 3 / June 2014
- Published online by Cambridge University Press:
- 22 August 2013, pp. 431-436
- Print publication:
- June 2014
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- Article
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- You have access
- Export citation
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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.
