Sós Permutations

Let for fixed real parameters α and β. For any positive integer n, define the Sós permutation π to be the lexicographically first permutation such that . In this article, we give a bijection between Sós permutations and regions in a partition of the parameter space . This allows us to enumerate these permutations and to obtain the following "three areas" theorem: in any vertical strip , with a Farey interval, there are at most three distinct areas of regions, and one of these areas is the sum of the other two.