Counting symmetric bracelets

Y. Zelenyuk; Y. Zelenyuk

Counting symmetric bracelets

Authors

Y. Zelenyuk University of the Witwatersrand
Y. Zelenyuk

Keywords:

Necklace, bracelet, symmetry, coloring, finite cyclic group, dihedral group.

Abstract

$r$ -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

$\frac{1}{2}(r+1)r^\frac{n}{2}$ if

$n$ is even, and

$r^\frac{n+1}{2}$ if

$n$ is odd. DOI: 10.1017/S0004972713000701

Published

2014-03-25

Issue

Vol. 89 No. 3 (2014)

Section

Articles

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