Minimal complexity of equidistributed infinite permutations

TL;DR

This paper introduces a new class of equidistributed infinite permutations, showing that their minimal complexity is linear and characterizing them as Sturmian permutations, extending classical word complexity results.

Contribution

It establishes that equidistributed permutations have minimal complexity exactly equal to n, and characterizes these as Sturmian permutations, linking permutation complexity to Sturmian words.

Findings

Equidistributed permutations have minimal complexity p(n)=n.

Such permutations are characterized as Sturmian permutations.

The result extends classical complexity results from words to permutations.

Abstract

An infinite permutation is a linear ordering of the set of natural numbers. An infinite permutation can be defined by a sequence of real numbers where only the order of elements is taken into account. In the paper we investigate a new class of {\it equidistributed} infinite permutations, that is, infinite permutations which can be defined by equidistributed sequences. Similarly to infinite words, a complexity $p(n)$ of an infinite permutation is defined as a function counting the number of its subpermutations of length $n$. For infinite words, a classical result of Morse and Hedlund, 1938, states that if the complexity of an infinite word satisfies $p(n) \leq n$ for some $n$, then the word is ultimately periodic. Hence minimal complexity of aperiodic words is equal to $n+1$, and words with such complexity are called Sturmian. For infinite permutations this does not hold: There exist aperiodic permutations with complexity functions growing arbitrarily slowly, and hence there are no permutations of minimal complexity. We show that, unlike for permutations in general, the minimal complexity of an equidistributed permutation $\alpha$ is $p_{\alpha}(n)=n$. The class of equidistributed permutations of minimal complexity coincides with the class of so-called Sturmian permutations, directly related to Sturmian words.