Fine and Wilf’s Theorem for k-Abelian Periods
Two words u and v are k-abelian equivalent if they contain the same number of occurrences of each factor of length k and, moreover, start and end with a same factor of length k − 1, respectively. This leads to a hierarchy of equivalence relations on words which lie properly in between the equality and abelian equality.
The goal of this paper is to analyze Fine and Wilf’s periodicity theorem with respect to these equivalence relations. A crucial question here is to ask how far two “periodic” processes must coincide in order to guarantee a common “period”. Fine and Wilf’s theorem characterizes this for words. Recently, the same was done for abelian words. We show here that for k-abelian periods the situation resembles that of abelian words: In general, there are no bounds, but the cases when such bounds exist can be characterized. Moreover, in the cases when such bounds exist we give nontrivial upper bounds for these, as well as lower bounds for some cases. Only in quite rare cases (in particular for k = 2) we can show that our upper and lower bounds match.
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