On the binomial equivalence classes of finite words

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.