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Abstract

We observe that the necklace polynomials $M_d(x) = \frac {1}{d}\sum _{e\mid d}\mu (e)x^{d/e}$ are highly reducible over $\mathbb {Q}$ with many cyclotomic factors. Furthermore, the sequence $\Phi _d(x) - 1$ of shifted cyclotomic polynomials exhibits a qualitatively similar phenomenon, and it is often the case that $\Phi _m(x)$ divides both $M_d(x)$ and $\Phi _d(x) - 1$. We explain these cyclotomic factors of $M_d(x)$ and $\Phi _d(x) - 1$ in terms of what we call the $d$th necklace operator. Finally, we show how these cyclotomic factors correspond to certain hyperplane arrangements in finite abelian groups.