On cardinalities of k-abelian equivalence classes

Abstract

Two words u and v are k-abelian equivalent if for each word x of length at most k, x occurs equally many times as a factor in both u and v. The notion of k-abelian equivalence is an intermediate notion between the abelian equivalence and the equality of words. In this paper, we study the equivalence classes induced by the k-abelian equivalence, mainly focusing on the cardinalities of the classes. In particular, we are interested in the number of singleton k-abelian classes, i.e., classes containing only one element. We find a connection between the singleton classes and cycle decompositions of the de Bruijn graph. We show that the number of classes of words of length n containing one single element is of order O(n Nm(k−1)−1 ), where N m (l) = 1 l d|l ϕ(d)m l/d is the number of necklaces of length l over an m-ary alphabet. We conjecture that the upper bound is sharp. We also remark that, for k even and m = 2, the lower bound Ω(n Nm(k−1)−1 ) follows from an old conjecture on the existence of Gray codes for necklaces of odd length. We verify this conjecture for necklaces of length up to 15.

Regularity of k-Abelian Equivalence Classes of Fixed Cardinality

“…In [11], k-Abelian equivalence is characterized in terms of rewriting, namely by k-switching. For this we define the following.…”

“…Now R k is clearly symmetric, so that the reflexive and transitive closure R * k of R k is an equivalence relation on Σ * . This gives quite a different characterization of the relation ∼ k in terms of rewriting, see [11] for more details:…”

“…Continuing this line of thought, a formula for computing the size of a k-Abelian equivalence class represented by a given word is obtained in [11]. In the following, a rooted spanning tree with root v of a graph G is a spanning tree of G for which all edges are directed towards the root vertex v.…”

“…Such a research was initiated in [12], and later continued, e.g., in [4]. Other topics of the k-Abelian equivalence such as k-Abelian repetitions, k-Abelian palindromicity and k-Abelian singletons were studied in [8,10,11,14], respectively.…”

“…A characterization of the k-Abelian equivalence in terms of rewriting was obtained in [11]. This was based on the so-called k-switching lemma.…”