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Keywords:
combinatorics on words; generalized Thue-Morse word; factor frequency
combinatorics on words; generalized Thue-Morse word; factor frequency
Summary:
We describe factor frequencies of the generalized Thue-Morse word ${\mathbf t}_{b,m}$ defined for $b \ge 2,$ $m \ge 1,$ $b,m \in \mathbb N$, as the fixed point starting in $0$ of the morphism $$\varphi_{b,m}(k)=k(k+1)\dots(k+b-1),$$ where $k \in \{0,1,\dots, m-1\}$ and where the letters are expressed modulo $m$. We use the result of Frid [4] and the study of generalized Thue-Morse words by Starosta [6].
References:
[1] Allouche, J.-P., Shallit, J.: Sums of digits, overlaps, and palindromes. Discrete Math. Theoret. Comput. Sci. 4 (2000), 1–10. MR 1755723 | Zbl 1013.11004
[2] Balková, L.: Factor frequencies in languages invariant under symmetries preserving factor frequencies. Integers – Electronic Journal of Combinatorial Number Theory 12 (2012), A36.
[3] Dekking, M.: On the Thue-Morse measure. Acta Univ. Carolin. Math. Phys. 33 (1992), 35–40. MR 1287223 | Zbl 0790.11017
[4] Frid, A.: On the frequency of factors in a D0L word. J. Automata, Languages and Combinatorics 3 (1998), 29–41. MR 1663865 | Zbl 0912.68116
[5] Queffélec, M.: Substitution dynamical systems – Spectral analysis. Lecture Notes in Math. 1294 (1987). Zbl 1225.11001
[6] Starosta, Š.: Generalized Thue-Morse words and palindromic richness. Kybernetika 48 (2012), 3, 361–370.
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