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ON A PARTITION PROBLEM OF FINITE ABELIAN GROUPS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 92 / Issue 1 / August 2015
- Published online by Cambridge University Press:
- 29 April 2015, pp. 24-31
- Print publication:
- August 2015
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- Article
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Let
$G$ be a finite abelian group and 
$A\subseteq G$. For 
$n\in G$, denote by 
$r_{A}(n)$ the number of ordered pairs 
$(a_{1},a_{2})\in A^{2}$ such that 
$a_{1}+a_{2}=n$. Among other things, we prove that for any odd number 
$t\geq 3$, it is not possible to partition 
$G$ into 
$t$ disjoint sets 
$A_{1},A_{2},\dots ,A_{t}$ with 
$r_{A_{1}}=r_{A_{2}}=\cdots =r_{A_{t}}$.
COUNTING SYMMETRIC COLOURINGS OF THE VERTICES OF A REGULAR POLYGON
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 90 / Issue 1 / August 2014
- Published online by Cambridge University Press:
- 10 January 2014, pp. 1-8
- Print publication:
- August 2014
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- Article
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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.
