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Symmetric colorings of $\mathbb{Z}_p^n$ |
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PP: 1009-1011 |
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doi:10.18576/amis/170607
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Author(s) |
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Yuliya Zelenyuk,
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Abstract |
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| Symmetries on a group $G$ are the mappings $G\ni x\mapsto gx^{-1}g\in G$, where $g\in G$. A coloring $\chi:G\to\{1,\ldots,r\}$ of $G$ is symmetric if it is invariant under some symmetry. We count the number $S_r(\mathbb{Z}_p^n)$ of symmetric $r$-colorings of $\mathbb{Z}_p^n$, the direct product of $n$ copies of the cyclic group of prime order $p$. As a consequence we obtain that $S_r(\mathbb{Z}_p^n)=p^nr^{\frac{p^n+1}{2}}+S_r(\mathbb{Z}_p^{n-1})$. |
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