Counting symmetric bracelets

Authors

  • Y. Zelenyuk University of the Witwatersrand
  • Y. Zelenyuk

Keywords:

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

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