Palindromic Ziv–Lempel and Crochemore factorizations of m-bonacci infinite words

Abstract

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

Closed Ziv-Lempel factorization of the $m$-bonacci words

“…In this section, we link two kinds of factorizations of the m-bonacci words, namely the palindromic and closed z-factorizations. In [13], we introduced a variation of the z-factorization, the palindromic z-factorization, in which each factor is palindromic. Also, we computed this factorization for the Fibonacci word and more generally for the m-bonacci words.…”

“…Ghareghani et al [12] determined z-factorizations for standard episturmian words. We introduced the palindromic z-factorizations by requiring each factor to be a palindrome and computed this factorization for the m-bonacci words [13]. In this work, based on the notion of closed words, which appeared in [5], we introduce the closed z-factorization and apply it to the infinite Fibonacci word and then to all m-bonacci words, for m > 2.…”

Kernel Words and Gap Sequences of the Tribonacci Word on an Infinite Alphabet

“…It is recognized that the tribonacci sequence augments the classical Fibonacci sequence from its binary form to a three-letter configuration. Beyond this, extensions involving k letters gave birth to the k-bonacci sequence, a topic that has witnessed rigorous explorations [9][10][11]. Notably, recent contributions by Ghareghani, Mohammad-Noori, and Sharifani [12,13] presented a broadened scope by generalizing the k-bonacci sequence to an infinite alphabet.…”

Some Properties of the Quasi-Tribonacci Sequence

“…Břinda, Pelantová and Turek [18] gave the extension of the Fibonacci sequence to an alphabet of m letters, which is called m-bonacci sequence. Jahannia, Mohammad-Noori, Rampersad and Stipulanti [19,20] studied the Ziv-Lempel factorization of m-bonacci words. Recently, Zhang, Wen and Wu [21] gave the extension of the Fibonacci sequence to the infinite alphabet N and studied its combinatorial properties, including the growth order, digit sum and several decompositions.…”