Abelian Square-Free Partial Words

@inproceedings{BlanchetSadri2010AbelianSP,
  title={Abelian Square-Free Partial Words},
  author={Francine Blanchet-Sadri and Jane I. Kim and Robert Mercas and William Severa and Sean Simmons},
  booktitle={LATA},
  year={2010}
}
Erdos raised the question whether there exist infinite abelian square-free words over a given alphabet (words in which no two adjacent subwords are permutations of each other). Infinite abelian square-free words have been constructed over alphabets of sizes as small as four. In this paper, we investigate the problem of avoiding abelian squares in partial words (sequences that may contain some holes). In particular, we give lower and upper bounds for the number of letters needed to construct… 
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References

SHOWING 1-10 OF 28 REFERENCES
Maximal abelian square-free words of short length
Freeness of partial words
Abelian Squares are Avoidable on 4 Letters
TLDR
It is proved that the morphism g itself is a-2-free, that is, g(w) is an a- 2-free word w in Σ*.
New Abelian Square-Free DT0L-Languages over 4 Letters
In 1906 Axel Thue [34] started the systematic study of structures in words. Consequently, he studied basic objects of theoretical computer science long before the invention of the computer or DNA. In
An Answer to a Conjecture on Overlaps in Partial Words Using Periodicity Algorithms
TLDR
An algorithm is proposed that produces an infinite word over a five-letter alphabet that is overlap-free even after the insertion of an arbitrary number of holes, answering affirmatively a conjecture from Blanchet-Sadri, Mercas, and Scott.
Algorithmic Combinatorics on Partial Words
TLDR
This paper focuses on two areas of algorithmic combinatorics on partial words, namely, pattern avoidance and subword complexity, and discusses recent contributions as well as a number of open problems.
On the Maximal Number of Cubic Runs in a String
TLDR
C cubic runs are investigated, in which the shortest period p satisfies 3p≤|v|, and the upper bound of 0.5 n is shown on the maximal number of such runs in a string of length n, and an infinite sequence of words over binary alphabet is constructed.
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