Generalized Thue-Morse words and palindromic richness
Kybernetika, Tome 48 (2012) no. 3, pp. 361-370
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library
We prove that the generalized Thue-Morse word $\mathbf{t}_{b,m}$ defined for $b \ge 2$ and $m \ge 1$ as $\mathbf{t}_{b,m} = \left ( s_b(n) \mod m \right )_{n=0}^{+\infty}$, where $s_b(n)$ denotes the sum of digits in the base-$b$ representation of the integer $n$, has its language closed under all elements of a group $D_m$ isomorphic to the dihedral group of order $2m$ consisting of morphisms and antimorphisms. Considering antimorphisms $\Theta \in D_m$, we show that $\mathbf{t}_{b,m}$ is saturated by $\Theta$-palindromes up to the highest possible level. Using the generalisation of palindromic richness recently introduced by the author and E. Pelantová, we show that $\mathbf{t}_{b,m}$ is $D_m$-rich. We also calculate the factor complexity of $\mathbf{t}_{b,m}$.
We prove that the generalized Thue-Morse word $\mathbf{t}_{b,m}$ defined for $b \ge 2$ and $m \ge 1$ as $\mathbf{t}_{b,m} = \left ( s_b(n) \mod m \right )_{n=0}^{+\infty}$, where $s_b(n)$ denotes the sum of digits in the base-$b$ representation of the integer $n$, has its language closed under all elements of a group $D_m$ isomorphic to the dihedral group of order $2m$ consisting of morphisms and antimorphisms. Considering antimorphisms $\Theta \in D_m$, we show that $\mathbf{t}_{b,m}$ is saturated by $\Theta$-palindromes up to the highest possible level. Using the generalisation of palindromic richness recently introduced by the author and E. Pelantová, we show that $\mathbf{t}_{b,m}$ is $D_m$-rich. We also calculate the factor complexity of $\mathbf{t}_{b,m}$.
@article{KYB_2012_48_3_a1,
author = {Starosta, \v{S}t\v{e}p\'an},
title = {Generalized {Thue-Morse} words and palindromic richness},
journal = {Kybernetika},
pages = {361--370},
year = {2012},
volume = {48},
number = {3},
mrnumber = {2975794},
language = {en},
url = {http://geodesic.mathdoc.fr/item/KYB_2012_48_3_a1/}
}
Starosta, Štěpán. Generalized Thue-Morse words and palindromic richness. Kybernetika, Tome 48 (2012) no. 3, pp. 361-370. http://geodesic.mathdoc.fr/item/KYB_2012_48_3_a1/