A k-abelian cube is a word uvw, where the factors u, v, and w are either pairwise equal, or have the same multiplicities for every one of their factors of length at most k. Previously it has been shown that k-abelian cubes are avoidable over a binary alphabet for k ≥ 8. Here it is proved that this holds for k ≥ 5.
@article{ITA_2014__48_4_467_0,
author = {Merca\c s, Robert and Saarela, Aleksi},
title = {5-abelian cubes are avoidable on binary alphabets},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
volume = {48},
year = {2014},
pages = {467-478},
doi = {10.1051/ita/2014020},
mrnumber = {3302498},
zbl = {1302.68229},
language = {en},
url = {http://dml.mathdoc.fr/item/ITA_2014__48_4_467_0}
}
Mercaş, Robert; Saarela, Aleksi. 5-abelian cubes are avoidable on binary alphabets. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 48 (2014) pp. 467-478. doi : 10.1051/ita/2014020. http://gdmltest.u-ga.fr/item/ITA_2014__48_4_467_0/
[1] , , , , and , Avoiding abelian squares in partial words. J. Combin. Theory Ser. A 119 (2012) 257-270. | MR 2844095 | Zbl 1233.68183
[2] , G. Scott, A generalization of Thue freeness for partial words. Theoret. Comput. Sci. 410 (2009) 793-800. | MR 2492017 | Zbl 1162.68029
[3] , and , Unary pattern avoidance in partial words dense with holes. In Proc. of Language and Automata Theory and Applications - 5th International Conference, LATA 2011, Tarragona, Spain, May 26-31, 2011. Edited by A.H. Dediu, S. Inenaga and C. Martín-Vide. Vol. 6638 of Lect. Notes Comput. Science. Springer (2011) 155-166. | MR 2893673 | Zbl pre05903561
[4] , Strongly nonrepetitive sequences and progression-free sets. J. Combin. Theory Ser. A 27 (1979) 181-185. | MR 542527 | Zbl 0437.05011
[5] , Some unsolved problems. Magyar Tudományos Akadémia Matematikai Kutató Intézete 6 (1961) 221-254. | MR 177846 | Zbl 0100.02001
[6] , Existence of an infinite ternary 64-abelian square-free word. RAIRO: ITA 48 (2014) 307-314. | Numdam | MR 3302490 | Zbl 1297.68192
[7] and , On the unavoidability of k-abelian squares in pure morphic words. J. Integer Seq. 16 (2013). | MR 3032392 | Zbl 1285.68135
[8] , and , Problems in between words and abelian words: k-abelian avoidability. Theoret. Comput. Sci. 454 (2012) 172-177. | MR 2966632 | Zbl 1280.68149
[9] , , and , Local squares, periodicity and finite automata. In Rainbow of Computer Science, edited by C. Calude, G. Rozenberg and A. Salomaa. Springer (2011) 90-101. | MR 2805552 | Zbl pre05900368
[10] , and , Fine and Wilf's theorem for k-abelian periods. Internat. J. Found. Comput. Sci. 24 (2013) 1135-1152. | MR 3189714 | Zbl 1293.68210
[11] , Abelian squares are avoidable on 4 letters. In Proc. of the 19th International Colloquium on Automata, Languages and Programming (1992) 41-52. | MR 1250629
[12] and , Freeness of partial words. Theoret. Comput. Sci. 389 (2007) 265-277. | MR 2363377 | Zbl 1154.68457
[13] A. Saarela, 3-abelian cubes are avoidable on binary alphabets. In Proc. of the 17th International Conference on Developments in Language Theory, edited by M.-P. Béal and O. Carton, vol. 7907 of Lect. Notes Comput. Science. Springer (2013) 374-383. | MR 3097343 | Zbl pre06182460
[14] , On some generalizations of abelian power avoidability. preprint (2013).
[15] , Über unendliche Zeichenreihen. Norske Vid. Selsk. Skr. I, Mat. Nat. Kl. Christiania 7 (1906) 1-22. (Reprinted in Selected Mathematical Papers of A. Thue, T. Nagell, editor, Universitetsforlaget, Oslo, Norway (1977) 139-158). | JFM 39.0283.01
[16] , Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske Vid. Selsk. Skr. I, Mat. Nat. Kl. Christiania 1 (1912) 1-67. (Reprinted in Selected Mathematical Papers of Axel Thue, T. Nagell, editor, Universitetsforlaget, Oslo, Norway (1977) 413-478). | JFM 44.0462.01