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arXiv · 2206.00995

On the Lie complexity of Sturmian words

Alessandro De Luca·Gabriele Fici
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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.

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Alessandro De Luca, Gabriele Fici. 2022-06-02. On the Lie complexity of Sturmian words. https://doi.org/10.1016/j.tcs.2022.10.009

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