For a given language L, we study the languages X such that for all distinct words u, v ∈ L, there exists a word x ∈ X that appears a different number of times as a factor in u and in v. In particular, we are interested in the following question: For which languages L does there exist a finite language X satisfying the above condition? We answer this question for all regular languages and for all sets of factors of infinite words.
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Two words u and v are said to be k-abelian equivalent if, for each word x of length at most k, the number of occurrences of x as a factor of u is the same as for v. We study some combinatorial properties of k-abelian equivalence classes. Our starting point is a characterization of k-abelian equivalence by rewriting, so-called k-switching. Using this characterization we show that, over any fixed alphabet, the language of lexicographically least representatives of k-abelian equivalence classes is a regular language. From this we infer that the sequence of the numbers of equivalence classes is ℕ-rational. Furthermore, we show that the above sequence is asymptotically equal to a certain polynomial depending on k and the alphabet size.
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