We introduce a variation of the Ziv-Lempel and Crochemore factorizations of words by requiring each factor to be a palindrome. We compute these factorizations for the Fibonacci word, and more generally, for all m-bonacci words.2010 Mathematics Subject Classification: 68R15.The Ziv-Lempel [9] and Crochemore [4] factorizations are two well-known factorizations of words used in text compression and other text algorithms. Here we apply them to infinite words. Let |u| denote the length of a finite word u. In this paper, we start indexing words at 0, i.e., if u is a finite word over the alphabet A, then we write u = u 0 · · · u |u|−1 where u i ∈ A for all 0 ≤ i < |u|. If w is an infinite word and u is a finite word, we say there is an occurrence of u at position j in w if w = puw ′ for some word p of length j and some infinite word w ′ . Given an infinite word w, the Ziv-Lempel or z-factorization of w is the factorization z(w) = (z 1
A word w is said to be closed if it has a proper factor x which occurs exactly twice in w, as a prefix and as a suffix of w. Based on the concept of Ziv-Lempel factorization, we define the closed z-factorization of finite and infinite words. Then we find the closed z-factorization of the infinite m-bonacci words for all m ≥ 2. We also classify closed prefixes of the infinite m-bonacci words.
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