S\'os Permutations

TL;DR

This paper studies Sós permutations generated by a linear mod 1 function, establishing a bijection with parameter space regions, enabling enumeration and revealing a three-area structure within Farey intervals.

Contribution

It introduces a bijection between Sós permutations and parameter space regions, allowing enumeration and analysis of their geometric structure.

Findings

Enumerates Sós permutations via parameter space regions.

Establishes a three-area theorem within Farey intervals.

Provides a combinatorial-geometric framework for these permutations.

Abstract

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.