More properties of the Fibonacci word on an infinite alphabet

Recently the Fibonacci word $W$ on an infinite alphabet was introduced by [Zhang et al., Electronic J. Combinatorics 24-2 (2017) #P2.52] as a fixed point of the morphism $\phi: (2i) \mapsto (2i)(2i+ 1),\ (2i+ 1) \mapsto (2i+ 2)$ over all $i \in \mathbb{N}$. In this paper we investigate the occurrence of squares, palindromes, and Lyndon factors in this infinite word.

[1]  Gregory Kucherov,et al.  On Maximal Repetitions in Words , 1999, FCT.

[2]  M. Lothaire.  Algebraic Combinatorics on Words , 2002 .

[3]  William F. Smyth,et al.  A taxonomy of suffix array construction algorithms , 2007, CSUR.

[4]  Kalle Saari.  Lyndon words and Fibonacci numbers , 2014, J. Comb. Theory, Ser. A.

[5]  Frantisek Franek,et al.  Algorithms to Compute the Lyndon Array , 2016, Stringology.

[6]  Costas S. Iliopoulos,et al.  A Characterization of the Squares in a Fibonacci String , 1997, Theor. Comput. Sci..

[7]  Giuseppe Pirillo,et al.  Episturmian words and some constructions of de Luca and Rauzy , 2001, Theor. Comput. Sci..

[8]  Zhi-Xiong Wen,et al.  Some Properties of the Fibonacci Sequence on an Infinite Alphabet , 2017, Electron. J. Comb..

[9]  Jamie Simpson,et al.  The Exact Number of Squares in Fibonacci Words , 1999, Theor. Comput. Sci..

[10]  Kazuya Tsuruta,et al.  The "Runs" Theorem , 2014, SIAM J. Comput..

[11]  Christophe Reutenauer,et al.  Lyndon words, permutations and trees , 2003, Theor. Comput. Sci..

[12]  Michael G. Main,et al.  Detecting leftmost maximal periodicities , 1989, Discret. Appl. Math..

[13]  Uwe Baier.  Linear-time Suffix Sorting - A New Approach for Suffix Array Construction , 2016, CPM.

[14]  V. Hoggatt.  Fibonacci and Lucas Numbers , 2020, Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science.