Geodesic
On $k$-abelian avoidability
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part IV, Tome 402 (2012), pp. 170-182

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We consider a recently defined notion of $k$-abelian equivalence of words by giving some basic results and concentrating on avoidability problems. This equivalence relation counts the numbers of factors of length $k$ for a fixed natural number $k$. We ask for the size of the smallest alphabet for which $k$-abelian squares and cubes can be avoided, respectively. For $2$-abelian squares this is four – as in the case of abelian words, while for $2$-abelian cubes we have only strong evidence that the size is two – as it is in the case of words. In addition, we point out a few properties of morphisms supporting the view that it might be difficult to find solutions to our questions by simply iterating a morphism. 
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     author = {M. Huova and J. Karhum\"aki},
     title = {On $k$-abelian avoidability},
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     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_402_a9/}
}
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M. Huova; J. Karhumäki. On $k$-abelian avoidability. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part IV, Tome 402 (2012), pp. 170-182. http://geodesic.mathdoc.fr/item/ZNSL_2012_402_a9/