Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-19T13:56:13.104Z Has data issue: false hasContentIssue false
  • English
  • Français

The number of binary rotation words∗

Published online by Cambridge University Press:  11 August 2014

A. Frid  and
D. Jamet
A. Frid
Affiliation:
Sobolev Institute of Mathematics SB RAS, Koptyug Av. 4, 630090, Novosibirsk, Russia, and Université de Lorraine, 34 cours Léopold, CS 25233, 54052 Nancy cedex, France.. anna.e.frid@gmail.com
D. Jamet
Affiliation:
LORIA, UMR 7503, Campus scientifique BP 239, 54506 Vandoeuvre-lès-Nancy cedex, France.
Get access

Abstract

We consider binary rotation words generated by partitions of the unit circle to two intervals and give a precise formula for the number of such words of length n. We also give the precise asymptotics for it, which happens to be Θ(n4). The result continues the line initiated by the formula for the number of all Sturmian words obtained by Lipatov [Problemy Kibernet. 39 (1982) 67–84], then independently by Mignosi [Theoret. Comput. Sci. 82 (1991) 71–84], and others.

Keywords

Rotationrotation wordsSturmian wordssubword complexitytotal complexity
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 48 , Issue 4: Special issue in the honor of the 14th "Journées Montoises d'Informatique Théorique" , October 2014 , pp. 453 - 465
Copyright
© EDP Sciences 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ambrož, P., Frid, A., Masáková, Z. and Pelantová, E., On the number of factors in codings of three interval exchange, Discr. Math. Theoret. Comput. Sci. 13 (2011) 51–66. Google Scholar
Berenstein, C.A., Kanal, L.N., Lavine, D. and Olson, E.C., A geometric approach to subpixel registration accuracy. Comput. Vision Graph. 40 (1987) 334–360. Google Scholar
Berstel, J. and Pocchiola, M., A geometric proof of the enumeration formula for Sturmian words. Int. J. Algebra Comput. 3 (1993) 349–355. Google Scholar
Berstel, J. and Pocchiola, M., Random generation of finite Sturmian words. Discr. Math. 153 (1996) 29–39. Google Scholar
Berstel, J. and Vuillon, L., Coding rotations on intervals. Theoret. Comput. Sci. 281 (2002) 99–107. Google Scholar
Cassaigne, J. and Frid, A.E., On the arithmetical complexity of Sturmian words. Theoret. Comput. Sci. 380 (2007) 304–316. Google Scholar
Frid, A., A lower bound for the arithmetical complexity of Sturmian words, Siberian Electron. Math. Rep. 2 (2005) 14–22 (in Russian, English abstract). Google Scholar
Lipatov, E.P., A classification of binary collections and properties of homogeneity classes. Problemy Kibernet. 39 (1982) 67–84 (in Russian). Google Scholar
M. Lothaire, Algebraic Combinatorics on Words. Cambridge University Press, Cambridge (2002).
Mignosi, F., On the number of factors of Sturmian words. Theoret. Comput. Sci. 82 (1991) 71–84. Google Scholar
Rote, G., Sequences with subword complexity 2n, J. Number Theory 46 (1994) 196–213. Google Scholar