Symmetric colorings of $\mathbb{Z}_p^n$

Authors

Yuliya Zelenyuk, School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South AfricaSchool of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South AfricaFollow

Author Country (or Countries)

South Africa

Abstract

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 $S_r(\mathbb{Z}_p^n)=p^nr^{\frac{p^n+1}{2}}+S_r(\mathbb{Z}_p^{n-1})$.

Digital Object Identifier (DOI)

https://dx.doi.org/10.18576/amis/170607

Recommended Citation

Zelenyuk, Yuliya (2023) "Symmetric colorings of $\mathbb{Z}_p^n$," Applied Mathematics & Information Sciences: Vol. 17: Iss. 6, Article 13.
DOI: https://dx.doi.org/10.18576/amis/170607
Available at: https://digitalcommons.aaru.edu.jo/amis/vol17/iss6/13