Infinite permutations vs. infinite words

TL;DR

This paper compares properties of infinite words and infinite permutations, exploring their periodicity, complexity, automaticity, and generation by infinite words, highlighting similarities and differences in their behaviors.

Contribution

It introduces and analyzes the properties of infinite permutations, extending concepts from infinite words, and discusses their generation, complexity, and automatic properties.

Findings

Permutations exhibit both similarities and differences to infinite words in periodicity and complexity.

Infinite permutations generated by infinite words have distinct automatic properties.

The study opens new directions for research in permutation and word theory.

Abstract

I am going to compare well-known properties of infinite words with those of infinite permutations, a new object studied since middle 2000s. Basically, it was Sergey Avgustinovich who invented this notion, although in an early study by Davis et al. permutations appear in a very similar framework as early as in 1977. I am going to tell about periodicity of permutations, their complexity according to several definitions and their automatic properties, that is, about usual parameters of words, now extended to permutations and behaving sometimes similarly to those for words, sometimes not. Another series of results concerns permutations generated by infinite words and their properties. Although this direction of research is young, many people, including two other speakers of this meeting, have participated in it, and I believe that several more topics for further study are really promising.