Möbius Polynomials

Abstract

We introduce the Möbius polynomial $M_n(x)={\sum }_{d|n}\mu \left(\frac{n}{d}\right)x^d$, which gives the number of aperiodic bracelets of length $n$ with $x$ possible types of gems, and therefore satisfies $M_n(x)\equiv 0$ (mod $n$) for all $x\in \mathbb{Z}$. We derive some key properties, analyze graphs in the complex plane, and then apply Möbius polynomials combinatorially to juggling patterns, irreducible polynomials over finite fields, and Euler's totient theorem.

10 pages, 2 figures