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On the Unavoidability of ***k*-Abelian Squares in Pure Morphic Words

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Mari Huova and Juhani Karhumäki

Department of Mathematics and Statistics

TUCS

University of Turku

20014 Turku

Finland

**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.

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(Concerned with sequences
A000002
A006156
A010060.)

Received July 1 2012;
revised version received October 11 2012.
Published in *Journal of Integer Sequences*, March 2 2013.

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