  Two finite words $u$ and $v$ are $k$-binomially equivalent if, for each word
$x$ of length at most $k$, $x$ appears the same number of times as a
subsequence (i.e., as a scattered subword) of both $u$ and $v$. This notion
generalizes abelian equivalence. In this paper, we study the equivalence
classes induced by the $k$-binomial equivalence with a special focus on the
cardinalities of the classes. We provide an algorithm generating the
$2$-binomial equivalence class of a word. For $k \geq 2$ and alphabet of $3$ or
more symbols, the language made of lexicographically least elements of every
$k$-binomial equivalence class and the language of singletons, i.e., the words
whose $k$-binomial equivalence class is restricted to a single element, are
shown to be non context-free. As a consequence of our discussions, we also
prove that the submonoid generated by the generators of the free nil-$2$ group
on $m$ generators is isomorphic to the quotient of the free monoid $\{ 1,
\ldots , m\}^{*}$ by the $2$-binomial equivalence.
