On the Lie complexity of Sturmian words

TL;DR

This paper provides an elementary combinatorial proof and an exact formula for the Lie complexity of Sturmian words, which are characterized by their minimal factor complexity, expanding understanding of their algebraic and combinatorial properties.

Contribution

It offers a new combinatorial proof and precise formula for the Lie complexity of Sturmian words, previously established through algebraic methods.

Findings

Lie complexity of Sturmian words is exactly 2 for all lengths

Elementary combinatorial proof confirms previous algebraic results

Exact formula for Lie complexity of any Sturmian word

Abstract

Bell and Shallit recently introduced the Lie complexity of an infinite word $s$ as the function counting for each length the number of conjugacy classes of words whose elements are all factors of $s$. They proved, using algebraic techniques, that the Lie complexity is bounded above by the first difference of the factor complexity plus one; hence, it is uniformly bounded for words with linear factor complexity, and, in particular, it is at most 2 for Sturmian words, which are precisely the words with factor complexity $n+1$ for every $n$. In this note, we provide an elementary combinatorial proof of the result of Bell and Shallit and give an exact formula for the Lie complexity of any Sturmian word.