Long aa2f word over 3 letters / Veikko Keränen

An aa2f word over 3 letters of length 15796

A word (or string) is called aa2f (almost abelian square-free) if it does not contain an abelian square uv as a factor - with the exception that the shortest squares of period 1 (|u| = |v| = 1) are allowed. In other words, the repetitions xx and xxx, where x is a letter, are allowed to appear but all the other abelian squares should be avoided.

Originally, the aa2f problem on three letters was posed by Sami Mäkelä in his M.Sc. Thesis, Patterns in words (in Finnish), University of Turku, 2002.
The problem whether there is an infinite word avoiding abelian squares of period more than 1 over the ternary alphabet is still open.
However, a partial solution to this problem was given by M. Rao and M. Rosenfeld in 2015 in their paper "On Mäkelä’s Conjectures: deciding if a morphic word avoids long abelian-powers".
Indeed, they show that abelian squares of period more than 5 are avoidable on the ternary alphabet.

Below, a long aa2f word over 3 letters is represented.

The first word is of length 15796 and it was found with the aid of a parallelized JavaScript program written by Alexandr Gavrilenko in 2015.

The rest twelve words, each of length 13796, are obtained by taking the middle part of the first word, that is, cutting away the first 1000 and the last 1000 characters, and listing, in alphabetical order, all the six permutations and their mirror images regarding this middle part.

© 2017 V. Keränen 15796 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January 13, 2017

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