Arithmetic functions and fixed points of powers of permutations
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866916409889521664 |
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| author | Nathanson, Melvyn B. |
| author_facet | Nathanson, Melvyn B. |
| contents | Let $σ$ be a permutation of a nonempty finite or countably infinite set $X$ and let $F_X\left( σ^k\right)$ count the number of fixed points of the $k$th power of $σ$. This paper explains how the arithmetic function $k \mapsto \left(F_X\left( σ^k\right) \right)_{k=1}^{\infty}$ determines the conjugacy class of the permutation $σ$, constructs an algorithm to compute the conjugacy class from the fixed point counting function $F_X\left( σ^k\right)$, and describes the arithmetic functions that are fixed point counting functions of permutations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_04021 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Arithmetic functions and fixed points of powers of permutations Nathanson, Melvyn B. Combinatorics Group Theory Number Theory 11N56, 20B05, 20B07, 20B10, 20F69 Let $σ$ be a permutation of a nonempty finite or countably infinite set $X$ and let $F_X\left( σ^k\right)$ count the number of fixed points of the $k$th power of $σ$. This paper explains how the arithmetic function $k \mapsto \left(F_X\left( σ^k\right) \right)_{k=1}^{\infty}$ determines the conjugacy class of the permutation $σ$, constructs an algorithm to compute the conjugacy class from the fixed point counting function $F_X\left( σ^k\right)$, and describes the arithmetic functions that are fixed point counting functions of permutations. |
| title | Arithmetic functions and fixed points of powers of permutations |
| topic | Combinatorics Group Theory Number Theory 11N56, 20B05, 20B07, 20B10, 20F69 |
| url | https://arxiv.org/abs/2206.04021 |