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.

We introduce a square root map on Sturmian words and study its properties. Given a Sturmian word of slope α, there exists exactly six minimal squares in its language (a minimal square does not have a square as a proper prefix). A Sturmian word s of slope α can be written as a product of these six minimal squares:The square root of s is defined to be the wordThe main result of this paper is that that √ s is also a Sturmian word of slope α. Further, we characterize the Sturmian fixed points of the square root map, and we describe how to find the intercept of √ s and an occurrence of any prefix of √ s in s. Related to the square root map, we characterize the solutions of the word equation X 2 1 X 2 2 · · · X 2 n = (X 1 X 2 · · · X n ) 2 in the language of Sturmian words of slope α where the words X 2 i are minimal squares of slope α. We also study the square root map in a more general setting. We explicitly construct an infinite set of non-Sturmian fixed points of the square root map. We show that the subshifts Ω generated by these words have a curious property: for all w ∈ Ω either √ w ∈ Ω or √ w is periodic. In particular, the square root map can map an aperiodic word to a periodic word.

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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 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.

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