An infinite permutation α is a linear ordering of N. We study properties of infinite permutations analogous to those of infinite words, and show some resemblances and some differences between permutations and words. In this paper, we try to extend to permutations the notion of automaticity. As we shall show, the standard definitions which are equivalent in the case of words are not equivalent in the context of permutations. We investigate the relationships between these definitions and prove that they constitute a chain of inclusions. We also construct and study an automaton generating the Thue-Morse permutation.
Mots-clés : permutation, infinite permutation, ordering, infinite word, automatic word, automatic permutation, Thue-Morse word, Thue-Morse permutation
@article{ITA_2012__46_1_77_0,
author = {Frid, Anna and Zamboni, Luca},
title = {On automatic infinite permutations},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
pages = {77--85},
publisher = {EDP-Sciences},
volume = {46},
number = {1},
year = {2012},
doi = {10.1051/ita/2011129},
mrnumber = {2904962},
zbl = {1247.05007},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ita/2011129/}
}
TY - JOUR AU - Frid, Anna AU - Zamboni, Luca TI - On automatic infinite permutations JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications PY - 2012 SP - 77 EP - 85 VL - 46 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ita/2011129/ DO - 10.1051/ita/2011129 LA - en ID - ITA_2012__46_1_77_0 ER -
%0 Journal Article %A Frid, Anna %A Zamboni, Luca %T On automatic infinite permutations %J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications %D 2012 %P 77-85 %V 46 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ita/2011129/ %R 10.1051/ita/2011129 %G en %F ITA_2012__46_1_77_0
Frid, Anna; Zamboni, Luca. On automatic infinite permutations. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 46 (2012) no. 1, pp. 77-85. doi : 10.1051/ita/2011129. http://www.numdam.org/articles/10.1051/ita/2011129/
[1] and , Automatic sequences - theory, applications, generalizations. Cambridge University Press (2003). | MR | Zbl
[2] , and , Periodicity, repetitions, and orbits of an automatic sequence. Theoret. Comput. Sci. 410 (2009) 2795-2803. | MR | Zbl
[3] , , and , Infinite permutations of lowest maximal pattern complexity. Theoret. Comput. Sci. 412 (2011) 2911-2921. | MR | Zbl
[4] , and , Enumeration and Decidable Properties of Automatic Sequences, Lect. Notes Comput. Sci. 6795 (2011) 165-179. | MR | Zbl
[5] , , and , Suites algébriques, automates et substitutions. Bull. Soc. Math. France 108 (1980) 401-419. | EuDML | Numdam | MR | Zbl
[6] , Uniform tag sequences. Math. Syst. Theor. 6 (1972) 164-192. | MR | Zbl
[7] , , and , On permutations containing no long arithmetic progressions. Acta Arith. 34 (1977) 81-90. | EuDML | MR | Zbl
[8] , Automata, Languages, and Machines A. Academic Press (1974). | Zbl
[9] -Flaass and A.E. Frid, On periodicity and low complexity of infinite permutations. Eur. J. Comb. 28 (2007) 2106-2114. | MR | Zbl
[10] , On permutations generated by infinite binary words. Sib. Èlectron. Mat. Izv. 3 (2006) 304-311 (in Russian, English abstract). | EuDML | MR | Zbl
[11] , On an infinite permutation similar to the Thue-Morse word. Discrete Math. 309 (2009) 6641-6643. | MR | Zbl
[12] , On the permutations generated by Sturmian words. Sib. Math. J. 50 (2009) 674-680. | EuDML | MR | Zbl
[13] , On the infinite permutation generated by the period doubling word. Eur. J. Comb. 31 (2010) 368-378. | MR | Zbl
[14] , Permutation complexity of the Thue-Morse word. Adv. Appl. Math. 47 (2011) 309-329. | MR | Zbl
Cité par Sources :
