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On unavoidability of \(k\)-abelian squares in pure morphic words. (English) Zbl 1285.68135

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.

MSC:

68R15 Combinatorics on words
05A05 Permutations, words, matrices

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