Multiplicative and Exponential Variations of Orthomorphisms of Cyclic Groups
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2017
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| _version_ | 1866910222707064832 |
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| author | Chen, Evan |
| author_facet | Chen, Evan |
| contents | An orthomorphism is a permutation $σ$ of $\{1, \dots, n-1\}$ for which $x + σ(x) \mod n$ is also a permutation on $\{1, \dots, n-1\}$. Eberhard, Manners, Mrazović, showed that the number of such orthomorphisms is $(\sqrt{e} + o(1)) \cdot \frac{n!^2}{n^n}$ for odd $n$ and zero otherwise.
In this paper we prove two analogs of these results where $x+σ(x)$ is replaced by $x σ(x)$ (a "multiplicative orthomorphism") or with $x^{σ(x)}$ (an "exponential orthomorphism"). Namely, we show that no multiplicative orthomorphisms exist for $n > 2$ but that exponential orthomorphisms exist whenever $n$ is twice a prime $p$ such that $p-1$ is squarefree. In the latter case we then estimate the number of exponential orthomorphisms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1710_02734 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Multiplicative and Exponential Variations of Orthomorphisms of Cyclic Groups Chen, Evan Combinatorics An orthomorphism is a permutation $σ$ of $\{1, \dots, n-1\}$ for which $x + σ(x) \mod n$ is also a permutation on $\{1, \dots, n-1\}$. Eberhard, Manners, Mrazović, showed that the number of such orthomorphisms is $(\sqrt{e} + o(1)) \cdot \frac{n!^2}{n^n}$ for odd $n$ and zero otherwise. In this paper we prove two analogs of these results where $x+σ(x)$ is replaced by $x σ(x)$ (a "multiplicative orthomorphism") or with $x^{σ(x)}$ (an "exponential orthomorphism"). Namely, we show that no multiplicative orthomorphisms exist for $n > 2$ but that exponential orthomorphisms exist whenever $n$ is twice a prime $p$ such that $p-1$ is squarefree. In the latter case we then estimate the number of exponential orthomorphisms. |
| title | Multiplicative and Exponential Variations of Orthomorphisms of Cyclic Groups |
| topic | Combinatorics |
| url | https://arxiv.org/abs/1710.02734 |