Berstel, Jean; Pocchiola, Michel A geometric proof of the enumeration formula for Sturmian words. (English) Zbl 0802.68099 Int. J. Algebra Comput. 3, No. 3, 349-355 (1993). The number of factors of length \(m\) of Sturmian words is known to be \(1 + \sum^ m_ 1 (m - i + 1) \varphi (i)\) where \(\varphi\) is the Euler function, i.e., \(\varphi (n)\) is the number of natural integers less than \(n\) and coprime to \(n\). We give a geometric proof of this formula, based on duality and on Euler’s relation for planar graphs. Reviewer: M.Pocchiola (Paris) Cited in 21 Documents MSC: 68R15 Combinatorics on words 68U05 Computer graphics; computational geometry (digital and algorithmic aspects) 68Q45 Formal languages and automata Keywords:combinatorics on words; infinite words; Sturmian words PDF BibTeX XML Cite \textit{J. Berstel} and \textit{M. Pocchiola}, Int. J. Algebra Comput. 3, No. 3, 349--355 (1993; Zbl 0802.68099) Full Text: DOI