Huova, Mari; Karhumäki, Juhani On unavoidability of \(k\)-abelian squares in pure morphic words. (English) Zbl 1285.68135 J. Integer Seq. 16, No. 2, Article 13.2.9, 11 p. (2013). Authors’ abstract: We consider a recently defined notion of \(k\)-abelian equivalence of words by concentrating on avoidability problems. The equivalence class of a word depends on the number of occurrences of different factors of length \(k\) for a fixed natural number \(k\) and the prefix of the word. We show that over a ternary alphabet, \(k\)-abelian squares cannot be avoided in pure morphic words for any natural number \(k\). Nevertheless, computational experiments support the conjecture that even \(3\)-abelian squares can be avoided over a ternary alphabet. This illustrates that the simple but widely used method to produce infinite words by iterating a single morphism is not powerful enough with \(k\)-abelian avoidability questions. Cited in 3 Documents MSC: 68R15 Combinatorics on words 05A05 Permutations, words, matrices Keywords:combinatorics on words; \(k\)-abelian equivalence; avoidability; pure morphic word Software:OEIS PDFBibTeX XMLCite \textit{M. Huova} and \textit{J. Karhumäki}, J. Integer Seq. 16, No. 2, Article 13.2.9, 11 p. (2013; Zbl 1285.68135) Full Text: EMIS Online Encyclopedia of Integer Sequences: Kolakoski sequence: a(n) is length of n-th run; a(1) = 1; sequence consists just of 1’s and 2’s. Number of ternary squarefree words of length n. Thue-Morse sequence: let A_k denote the first 2^k terms; then A_0 = 0 and for k >= 0, A_{k+1} = A_k B_k, where B_k is obtained from A_k by interchanging 0’s and 1’s.