Abstract
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. We show that the set of lexicographically least representatives of equivalence classes is a regular language. From this we infer that the sequence of the numbers of equivalence classes is \(\mathbb {N}\)-rational. We also show that the set of words defining k-abelian singleton classes is regular.
J. Karhumäki—Supported by the Academy of Finland, grant 257857.
S. Puzynina—Supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).
M.A. Whiteland—Supported by the Academy of Finland, grant 257857.
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Acknowledgments
The automata used to calculate the functions in Propositions 14 and 15 were constructed using the java package dk.brics.automaton [14]. The automata in Fig. 3 were created using the software Graphviz [4]. We would like to thank the anonymous referees for valuable comments which helped to improve the presentation.
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Cassaigne, J., Karhumäki, J., Puzynina, S., Whiteland, M.A. (2016). k-Abelian Equivalence and Rationality. In: Brlek, S., Reutenauer, C. (eds) Developments in Language Theory. DLT 2016. Lecture Notes in Computer Science(), vol 9840. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53132-7_7
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