Some Combinatorial Properties of Sturmian Words

Abstract In this paper we give a characterization of finite Sturmian words, by palindrome words, which generalizes a property of the Fibonacci words. We prove that the set St of finite Sturmian words coincides with the set of the factors of all the words w such that w = AB = Cxy with A, B, C palindromes, x , y ϵ{ a,b } and x ≠ y . Moreover, using this result we prove that St is equal to the set of the factors of all words w having two periods p and q which are coprimes and such that | w | ⩾ p + q − 2. Several other combinatorial properties concerning special and bispecial elements of St are shown. As a consequence we give a new, and purely combinatorial, proof of the enumeration formula of St .