  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 $\mathcal
O(n^{N_m(k-1)-1})$, where $N_m(l) = \tfrac{1}{l}\sum_{d\mid l}
\varphi(d)m^{l/d}$ 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 $\Omega(n^{N_m(k-1)-1})$ 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.
