k-Abelian Equivalence and Rationality

Abstract

<p>Two words $u$ and $v$ are said to be $k$-abelian equivalent if, for each word $x$ of length at<br />most $k$, the number of occurrences of $x$ as a factor of $u$ is the same as for $v$. We study<br />some combinatorial properties of $k$-abelian equivalence classes. Our starting point is a<br />characterization of $k$-abelian equivalence by rewriting, so-called $k$-switching. We show that<br />the set of lexicographically least representatives of equivalence classes is a regular language.<br />From this we infer that the sequence of the numbers of equivalence classes is $N$-rational. We<br />also show that the set of words defining $k$-abelian singleton classes is regular.<br /></p>