  Let $f(x) = \alpha x + \beta \mod 1$ for fixed real parameters $\alpha$ and
$\beta$. For any positive integer $n$, define the S\'os permutation $\pi$ to be
the lexicographically first permutation such that $0 \leq f(\pi(0)) \leq
f(\pi(1)) \leq \cdots \leq f(\pi(n)) < 1$. In this article we give a bijection
between S\'os permutations and regions in a partition of the parameter space
$(\alpha,\beta)\in [0,1)^2$. This allows us to enumerate these permutations and
to obtain the following "three areas" theorem: in any vertical strip
$(a/b,c/d)\times [0,1)$, with $(a/b,c/d)$ a Farey interval, there are at most
three distinct areas of regions, and one of these areas is the sum of the other
two.
