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On automatic infinite permutations∗

Published online by Cambridge University Press:  23 November 2011

Anna Frid  and
Luca Zamboni
Anna Frid
Affiliation:
Sobolev Institute of Mathematics SB RAS, Koptyug av. 4, 630090 Novosibirsk, Russia.. anna.e.frid@gmail.com Institut Camille Jordan, Université Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France.; zamboni@math.univ-lyon1.fr.
Luca Zamboni
Affiliation:
Institut Camille Jordan, Université Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France.; zamboni@math.univ-lyon1.fr. Department of Mathematics and Turku Centre for Computer Science, University of Turku, 20014 Turku, Finland.
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Abstract

An infinite permutation α is a linear ordering of N. We study propertiesof infinite permutations analogous to those of infinite words, and show some resemblancesand some differences between permutations and words. In this paper, we try to extend topermutations the notion of automaticity. As we shall show, the standard definitions whichare equivalent in the case of words are not equivalent in the context of permutations. Weinvestigate the relationships between these definitions and prove that they constitute achain of inclusions. We also construct and study an automaton generating the Thue-Morsepermutation.

Keywords

Permutationinfinite permutationorderinginfinite wordautomatic wordautomatic permutationThue-Morse wordThue-Morse permutation

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References

J.-P. Allouche and J. Shallit, Automatic sequences – theory, applications, generalizations. Cambridge University Press (2003).
Allouche, J.-P., Rampersad, N. and Shallit, J., Periodicity, repetitions, and orbits of an automatic sequence. Theoret. Comput. Sci. 410 (2009) 2795–2803. Google Scholar
Avgustinovich, S., Frid, A., Kamae, T. and Salimov, P., Infinite permutations of lowest maximal pattern complexity. Theoret. Comput. Sci. 412 (2011) 2911–2921. Google Scholar
Charlier, É., Rampersad, N. and Shallit, J., Enumeration and Decidable Properties of Automatic Sequences, Lect. Notes Comput. Sci. 6795 (2011) 165–179. Google Scholar
Christol, G., Kamae, T., France, M.M. and Rauzy, G., Suites algébriques, automates et substitutions. Bull. Soc. Math. France 108 (1980) 401–419. Google Scholar
Cobham, A., Uniform tag sequences. Math. Syst. Theor. 6 (1972) 164–192. Google Scholar
Davis, J.A., Entringer, R.C., Graham, R.L. and Simmons, G.J., On permutations containing no long arithmetic progressions. Acta Arith. 34 (1977) 81–90. Google Scholar
S. Eilenberg, Automata, Languages, and Machines A. Academic Press (1974).
D.G. Fon-Der-Flaass and Frid, A.E., On periodicity and low complexity of infinite permutations. Eur. J. Comb. 28 (2007) 2106–2114. Google Scholar
Makarov, M., On permutations generated by infinite binary words. Sib. Èlectron. Mat. Izv. 3 (2006) 304–311 (in Russian, English abstract). Google Scholar
Makarov, M., On an infinite permutation similar to the Thue-Morse word. Discrete Math. 309 (2009) 6641–6643. Google Scholar
Makarov, M., On the permutations generated by Sturmian words. Sib. Math. J. 50 (2009) 674–680. Google Scholar
Makarov, M., On the infinite permutation generated by the period doubling word. Eur. J. Comb. 31 (2010) 368–378. Google Scholar
Widmer, S., Permutation complexity of the Thue-Morse word. Adv. Appl. Math. 47 (2011) 309–329. Google Scholar