Abstract
In this paper, we study the Thue-Morse word on a ternary alphabet. We establish some properties on special factors of this word and prove that it is 2-balanced. Moreover, we determine its Abelian complexity function.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Allouche, J.P., Arnold, A., Berstel, J., Brlek, S., Jockush, W., Plouffe, S., Sagan, B.E.: A relative of the thue-morse sequence. Disc. Math. 139, 455–461 (1995)
Allouche, J.P., Peyrière, J., Wen, Z.-X., Wen, Z.-Y.: Hankel determinants of the thue-morse sequense. Ann. Inst. Fourier 48(1), 1–27 (1998)
Allouche, J.P., Shallit, J.: The Ubiquitous Prouhet-Thue-Morse Sequence. In: Ding, C., Helleseth, T., Niederreiter, H. (eds.) Sequences and their Applications. Discrete Mathematics and Theoretical Computer Science, pp. 1–16. Springer, London (1999)
Balkovà, L., Břinda, K., Turek, O.: Abelian complexity of infinite words associated with non-simple parry number. Theor. Comput. Sci. 412, 6252–6260 (2011)
Cassaigne, J.: Facteurs spéciaux. Bull. Belg. Math. 4, 67–88 (1997)
Cassaigne, J., Kaboré, I.: Abelian complexity and frequencies of letters in infinite words. Int. J. Found. Comput. Sci. 27(05), 631–649 (2016)
Cassaigne, J., Richomme, G., Saari, K., Zamboni, L.Q.: Avoiding abelian powers in binary wordswith bounded abelian complexity. Int. J. Found. Comput. Sci. 22(2011), 905–920 (2011)
Chen, J., Lü, X., Wu, W.: On the k-abelian complexity of the cantor sequence. arXiv: 1703.04063 (2017)
Coven, E.M., Hedlund, G.A.: Sequences with minimal block growth. Math. Syst. Theo. 7, 138–153 (1973)
Curie, J., Rampersad, N.: Recurrent words with constant abelian complexity. Adv. Appl. Math. 47, 116–124 (2011)
Gottschalk, W.H.: Substitution on minimal sets. Trans. Amer. Math. Soc. 109, 467–491 (1963)
Lü, X., Chen, J., Wen, Z., Wu, W.: On the abelian complexity of the rudin-shapiro sequence. J. Math. Anal. Appl. 451, 822–838 (2017)
Madill, B., Rampersad, N.: The abelian complexity of the paperfolding word. Disc. Math. 313, 831–838 (2013)
Massé, A.B., Brlek, S., Garon, A., Labbé, S.: Combinatorial properties of \(f\)-palindromes in the thue-morse sequence. Pure Math. Appl. 19, 39–52 (2008)
Parreau, A., Rigo, M., Rowland, E., Vandomme, E.: A new approach to the \(2-\)regularity of the \(l-\)abelian complexity of \(2-\)automatic sequences. Electron. J. Combin. 22, 1–27 (2015)
Richomme, G., Saari, K., Zamboni, L.Q.: Balance and abelian complexity of the tribonacci word. Adv. Appl. Math. 45, 212–231 (2010)
Richomme, G., Saari, K., Zamboni, L.Q.: Abelian complexity in minimal subshifts. J. London Math. Soc. 83, 79–95 (2011)
Štarosta, S.: Generalised thue-morse word and palindromic richness. Kybernetika 48(3), 361–370 (2012)
Thue, A.: Über unendliche zeichenreihen. Norske Vid. Selsk. Skr. Mat. Nat. Kl. 7, 1–22 (1906)
Thue, A.: Über die gegenseilige lage gleicher teile gewisser zeichenreihen. Norske Vid. Selsk. Skr. Mat. Nat. Kl. 1, 139–158 (1912)
Turek, O.: Balance and abelian complexity of a certain class of infinite ternary words. RAIRO Theor. Inf. Appl. 44, 313–337 (2010)
Turek, O.: Abelian complexity and abelian co-decomposition. Theor. Comput. Sci. 469, 77–91 (2013)
Turek, O.: Abelian complexity of the tribonacci word. J. Integer Seq. 18, 212–231 (2015)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Kaboré, I., Kientéga, B. (2017). Abelian Complexity of Thue-Morse Word over a Ternary Alphabet. In: Brlek, S., Dolce, F., Reutenauer, C., Vandomme, É. (eds) Combinatorics on Words. WORDS 2017. Lecture Notes in Computer Science(), vol 10432. Springer, Cham. https://doi.org/10.1007/978-3-319-66396-8_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-66396-8_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-66395-1
Online ISBN: 978-3-319-66396-8
eBook Packages: Computer ScienceComputer Science (R0)