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COUNTING SYMMETRIC COLOURINGS OF THE VERTICES OF A REGULAR POLYGON

Part of: Graph theory Elementary number 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 yuliya.zelenyuk@wits.ac.za
YULIYA ZELENYUK
Affiliation:
School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa email yuliya.zelenyuk@wits.ac.za
*
✉yevhen.zelenyuk@wits.ac.za
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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

Aigner, M., Combinatorial Theory (Springer, Berlin, 1979).CrossRefGoogle Scholar
Bender, E. and Goldman, J., ‘On the applications of Möbius inversion in combinatorial analysis’, Amer. Math. Monthly 82 (1975), 789–803.Google Scholar
Gryshko, Y., ‘Symmetric colorings of regular polygons’, Ars Combinatorica 78 (2006), 277–281.Google Scholar
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