A new geometric approach to Sturmian words
Kaisa Matom\"aki, Kalle Saari
TL;DR
This paper presents a novel geometric method to analyze Sturmian words, providing new proofs for their enumeration, palindromic properties, and return word structure, enhancing understanding of their combinatorial characteristics.
Contribution
It introduces a geometric mapping approach to Sturmian words, offering new proofs for key properties and enumeration formulas, advancing combinatorial analysis techniques.
Findings
New geometric proof for enumeration of Sturmian words
New proof for palindromic Sturmian words enumeration
Alternative proof for the return words property
Abstract
We introduce a new geometric approach to Sturmian words by means of a mapping that associates certain lines in the n x n -grid and sets of finite Sturmian words of length n. Using this mapping, we give new proofs of the formulas enumerating the finite Sturmian words and the palindromic finite Sturmian words of a given length. We also give a new proof for the well-known result that a factor of a Sturmian word has precisely two return words.
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