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COUNTING SYMMETRIC COLOURINGS OF THE VERTICES OF A REGULAR POLYGON

Part of: Elementary number theory Graph theory Combinatorics

Published online by Cambridge University Press:  10 January 2014

YEVHEN ZELENYUK  and
YULIYA ZELENYUK
YEVHEN ZELENYUK*
Affiliation:
School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa email [email protected]
YULIYA ZELENYUK
Affiliation:
School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa email [email protected]
*
✉[email protected]
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Abstract

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A colouring of the vertices of a regular polygon is symmetric if it is invariant under some reflection of the polygon. We count the number of symmetric $r$-colourings of the vertices of a regular $n$-gon.

Keywords

symmetric colouringregular polygonfinite Abelian groupMöbius functionmultiplicative function

MSC classification

Secondary: 05A15: Exact enumeration problems, generating functions 05C15: Coloring of graphs and hypergraphs 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) 11A25: Arithmetic functions; related numbers; inversion formulas
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 1 , August 2014 , pp. 1 - 8
Copyright
Copyright ©2014 Australian Mathematical Publishing Association Inc. 

References

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Vinogradov, I., Elements of Number Theory (Dover Publications, Mineola, NY, 1954).Google Scholar
Zelenyuk, Y. and Zelenyuk, Yu., ‘Counting symmetric bracelets’, Bull. Aust. Math. Soc., to appear. Published online 22 August 2013.Google Scholar