TL;DR
This paper characterizes bispecial Sturmian words as maximal internal factors of Christoffel words, linking combinatorics on words with geometric representations and providing enumeration formulas.
Contribution
It establishes a precise characterization of bispecial Sturmian words in terms of Christoffel words, extending known relations and enabling enumeration.
Findings
Bispecial Sturmian words are maximal internal factors of Christoffel words.
Provides an enumerative formula for bispecial Sturmian words.
Investigates minimal forbidden words for Sturmian words.
Abstract
A finite Sturmian word w over the alphabet {a,b} is left special (resp. right special) if aw and bw (resp. wa and wb) are both Sturmian words. A bispecial Sturmian word is a Sturmian word that is both left and right special. We show as a main result that bispecial Sturmian words are exactly the maximal internal factors of Christoffel words, that are words coding the digital approximations of segments in the Euclidean plane. This result is an extension of the known relation between central words and primitive Christoffel words. Our characterization allows us to give an enumerative formula for bispecial Sturmian words. We also investigate the minimal forbidden words for the set of Sturmian words.
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