An orthomorphism is a permutation $\sigma$ of $\{ 1, \dots, n-1\}$ for which $x + \sigma(x) \mod n$ is also a permutation on $\{ 1, \dots, n-1\}$. Eberhard, Manners, Mrazovi\'c, 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 $x+\sigma(x)$ is replaced by $x \sigma(x)$ (a "multiplicative orthomorphism") or with $x^{ \sigma(x)}$ (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.
Categories: math.CO