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.
Mots-clés : combinatorics on words, k-abelian equivalence, repetition-freeness, cube-freeness
@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}, pages = {467--478}, publisher = {EDP-Sciences}, volume = {48}, number = {4}, year = {2014}, doi = {10.1051/ita/2014020}, mrnumber = {3302498}, zbl = {1302.68229}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/ita/2014020/} }
TY - JOUR AU - Mercaş, Robert AU - Saarela, Aleksi TI - 5-abelian cubes are avoidable on binary alphabets JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications PY - 2014 SP - 467 EP - 478 VL - 48 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ita/2014020/ DO - 10.1051/ita/2014020 LA - en ID - ITA_2014__48_4_467_0 ER -
%0 Journal Article %A Mercaş, Robert %A Saarela, Aleksi %T 5-abelian cubes are avoidable on binary alphabets %J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications %D 2014 %P 467-478 %V 48 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ita/2014020/ %R 10.1051/ita/2014020 %G en %F 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) no. 4, pp. 467-478. doi : 10.1051/ita/2014020. http://archive.numdam.org/articles/10.1051/ita/2014020/
[1] Avoiding abelian squares in partial words. J. Combin. Theory Ser. A 119 (2012) 257-270. | MR | Zbl
, , , , and ,[2] G. Scott, A generalization of Thue freeness for partial words. Theoret. Comput. Sci. 410 (2009) 793-800. | MR | Zbl
,[3] 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
, and ,[4] Strongly nonrepetitive sequences and progression-free sets. J. Combin. Theory Ser. A 27 (1979) 181-185. | MR | Zbl
,[5] Some unsolved problems. Magyar Tudományos Akadémia Matematikai Kutató Intézete 6 (1961) 221-254. | MR | Zbl
,[6] Existence of an infinite ternary 64-abelian square-free word. RAIRO: ITA 48 (2014) 307-314. | Numdam | MR | Zbl
,[7] On the unavoidability of k-abelian squares in pure morphic words. J. Integer Seq. 16 (2013). | MR | Zbl
and ,[8] Problems in between words and abelian words: k-abelian avoidability. Theoret. Comput. Sci. 454 (2012) 172-177. | MR | Zbl
, and ,[9] 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
, , and ,[10] Fine and Wilf's theorem for k-abelian periods. Internat. J. Found. Comput. Sci. 24 (2013) 1135-1152. | MR | Zbl
, and ,[11] Abelian squares are avoidable on 4 letters. In Proc. of the 19th International Colloquium on Automata, Languages and Programming (1992) 41-52. | MR
,[12] Freeness of partial words. Theoret. Comput. Sci. 389 (2007) 265-277. | MR | Zbl
and ,[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
[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
,[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
,Cité par Sources :