On the number of factors of Sturmian words
Theoretical Computer Science
Some combinatorial properties of Sturmian words
Theoretical Computer Science
Sturmian words, Lyndon words and trees
Theoretical Computer Science
Theoretical Computer Science
Sturmian words: structure, combinatorics, and their arithmetics
Theoretical Computer Science - Special issue: formal language theory
Theory of Codes
Combinatories of Standard Sturmian Words
Structures in Logic and Computer Science, A Selection of Essays in Honor of Andrzej Ehrenfeucht
Harmonic and gold Sturmian words
European Journal of Combinatorics
Codes of central Sturmian words
Theoretical Computer Science - The art of theory
On an involution of Christoffel words and Sturmian morphisms
European Journal of Combinatorics
Involutions of epicentral words
European Journal of Combinatorics
A standard correspondence on epicentral words
European Journal of Combinatorics
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A word w is central if it has a minimal period πw such that |w|-πw+2 is a period of w coprime with πw. Central words are in a two-letter alphabet A and play an essential role in combinatorics of Sturmian words. We study some new structural properties of the set PER of central words which are based on the existence of two basic bijections ψ and ϕ of A* in PER, related to two different methods of generation, and two natural bijections θ (the ratio of periods) and η (the rate) of PER in the set of all positive irreducible fractions. In this paper we are mainly interested in sets of central words which are codes. In particular, for any positive integer n we consider the set Δn of all central words w such that the period |w|-πw+2 is not larger than n+1 and |w| ≥ n. In a previous paper we proved that for each n, Δn is a maximal prefix central code called the Farey code of order n since it is naturally associated with the Farey series of order n+1. New structural properties of Farey codes are given as well as of their pre-codes Pn. In particular one has PER=∪n≥0Δn. Moreover, for each n two languages of central words Ln and Mn are introduced. The language Ln (resp., Mn) is called the Farey (resp., dual Farey) language of order n. The name is motivated by the fact that Ln and Mn, give faithful representations of the set of Farey's fractions of order n. Finally, two total orderings of PER are naturally defined in terms of maps θ and η. The notion of order of a central word relative to a language of central words is given and some general results are proved. In the case of Farey's languages one has that the Riemann hypothesis on the Zeta function can be restated in terms of a combinatorial property of these languages.