Infinite permutations of lowest maximal pattern complexity

TL;DR

This paper explores the complexity of infinite permutations, establishing that minimal complexity permutations are constructed from Sturmian words and characterizing their properties.

Contribution

It introduces the concept of maximal pattern complexity for infinite permutations and characterizes non-periodic permutations with minimal complexity using Sturmian words.

Findings

Maximal pattern complexity is ultimately constant iff the permutation is ultimately periodic.

Non-periodic permutations with minimal complexity are constructed from Sturmian words.

The paper provides a complete characterization of permutations with minimal complexity.

Abstract

An infinite permutation is a linear ordering of the set of non-negative integers. Generally, the properties of infinite permutations analogous to those of infinite words show some resemblances and some differences between permutations and words. In this paper, we define maximal pattern complexity for infinite permutations and show that this complexity function is ultimately constant if and only if the permutation is ultimately periodic. Then we characterize the non-periodic permutations of minimal complexity (equal to n) and find that they all are constructed with the use of Sturmian words.