k-Abelian Equivalence and Rationality

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. Using this characterization 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 N-rational. Furthermore, we sharpen an earlier result by showing that the k-abelian complexity function is asymptotic to a polynomial which depends on k and the alphabet size.

Regularity of k-Abelian Equivalence Classes of Fixed Cardinality

“…From the above it was concluded in [2] that also the union of two-element equivalence classes constitutes a regular set. This is based on the closure properties of regular sets and the above-mentioned three regular sets: the singletons, the set of minimal, and maximal elements in the equivalence classes.…”

“…This turned out to be quite an interesting approach. It allowed, see [2,3], to show that the union of singleton classes is a regular language. Similarly, the set of all lexicographically least (or, equivalently, greatest) representatives constitutes a regular set.…”

“…This latter sequence can be viewed as the complexity function of the k-Abelian equivalence. For a few small values of k and alphabet size m, the sequences are computed in [2], and are included in the On-Line Encyclopedia of Integer Sequences [17] as the sequences A289657 and A289658.…”

“…In [2], the main usage of the tool of the regular languages was to show that the number of equivalence classes of length n is asymptotic to a certain polynomial. However, by using the ideas presented in [2], we are able to prove our main theorem. For each r ≥ 1, the union of the k-Abelian equivalence classes of cardinality r is a regular set.…”

Separating Many Words by Counting Occurrences of Factors

“…We are specifically interested in the case of regular languages. Some connections between k-abelian equivalence and regular languages have been studied by Cassaigne, Karhumäki, Puzynina and Whiteland [2]. Question 1.4.…”