Abelian-square factors and binary words
aa r X i v : . [ c s . F L ] J u l Abelian-square factors and binary words
Salah Triki
Mir@cl LaboratoryUniversity of Sfax, Tunisia [email protected]
Abstract.
In this work, we affirm the conjecture proposed by GabrieleFici and Filippo Mignosi in [1].
Keywords:
Abelian-square · Factor · Binary word
Definition 1.
A word w is called a f actor of a word u if there exists words x,y such that u = xwy . Definition 2.
An abelian-square is a word of length n where the first n symbolsform an anagram of the last n symbols. Lemma 1.
Let w is a word of length n , containing k many distinct abelian-square factors, and with the last symbol is in an abelian-square factor. Thena binary word of length n containing at least k many distinct abelian-squarefactors, and with the last symbol is in an abelian-square factor, exists. The binary word will be called a binary image of w . Proof.
By induction on n . For n < = 2, the claim is clear.Assume that the claim holds for a word w of length n and w ′ is a binary imageof w . So, wx with x equals to the last symbol of w , has k many distinct abelian-square factors, and has a length n + 1. And, w ′ y with y equals to the last symbolof w ′ is a binary image of wx ⊓⊔ Conjecture 1. [1] Assume that a word with length n , and containing k manydistinct abelian-square factors, exists. Then a binary word of length n containingat least k many distinct abelian-square factors exists. Proof.
By induction on n . For n < = 2, the claim is clear.Assume that the claim holds for a word w of length n . So, if the last symbol of w is in a factor, then by lemma 1, wx with x equals to the last symbol of w hasa binary image of length n + 1 with at least k distinct abelian-square factors. Ifthe last symbol is not in a factor, then also by lemma 1, wx has a binary imagewith at least k + 1 distinct abelian-square factors ⊓⊔ S. Triki
Acknowledgements
The author acknowledges, the financial support of this work from the TunisianGeneral Direction of Scientific Research (DGRST).
References
1. Fici, G., Mignosi, F.: Words with the maximum number of abelian squares. CoRR abs/1506.03562 (2015),(2015),