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arXiv · 2206.04021

Arithmetic functions and fixed points of powers of permutations

Melvyn B. Nathanson
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Abstract

Let $\sigma$ be a permutation of a nonempty finite or countably infinite set $X$ and let $F_X\left( \sigma^k\right)$ count the number of fixed points of the $k$th power of $\sigma$. This paper explains how the arithmetic function $k \mapsto \left(F_X\left( \sigma^k\right) \right)_{k=1}^{\infty}$ determines the conjugacy class of the permutation $\sigma$, constructs an algorithm to compute the conjugacy class from the fixed point counting function $F_X\left( \sigma^k\right)$, and describes the arithmetic functions that are fixed point counting functions of permutations.

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Melvyn B. Nathanson. 2022-06-08. Arithmetic functions and fixed points of powers of permutations. https://doi.org/10.1007/s00013-023-01855-0

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