Abelian maximal pattern complexity of words

TL;DR

This paper investigates the maximal pattern complexity of infinite words under Abelian equivalence, providing bounds and characterizations for binary recurrent aperiodic words.

Contribution

It introduces a lower bound for Abelian maximal pattern complexity and characterizes binary recurrent aperiodic words where this bound is achieved.

Findings

Lower bound for Abelian maximal pattern complexity established.

Bound is achieved for binary recurrent aperiodic words.

Characterization of such words based on the bound.

Abstract

In this paper we study the maximal pattern complexity of infinite words up to Abelian equivalence. We compute a lower bound for the Abelian maximal pattern complexity of infinite words which are both recurrent and aperiodic by projection. We show that in the case of binary words, the bound is actually achieved and gives a characterization of recurrent aperiodic words.