On the binomial equivalence classes of finite words
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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 ≥ 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, ... , m}^* by the 2-binomial equivalence.
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