Two finite words [Formula: see text] and [Formula: see text] are [Formula: see text]-binomially equivalent if, for each word [Formula: see text] of length at most [Formula: see text], [Formula: see text] appears the same number of times as a subsequence (i.e., as a scattered subword) of both [Formula: see text] and [Formula: see text]. This notion generalizes abelian equivalence. In this paper, we study the equivalence classes induced by the [Formula: see text]-binomial equivalence. We provide an algorithm generating the [Formula: see text]-binomial equivalence class of a word. For [Formula: see text] and alphabet of [Formula: see text] or more symbols, the language made of lexicographically least elements of every [Formula: see text]-binomial equivalence class and the language of singletons, i.e., the words whose [Formula: see text]-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-[Formula: see text] group (also called free nilpotent group of class [Formula: see text]) on [Formula: see text] generators is isomorphic to the quotient of the free monoid [Formula: see text] by the [Formula: see text]-binomial equivalence.

中文翻译：

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有限词的二项式等价类

两个有限词 [Formula: see text] 和 [Formula: see text] 是 [Formula: see text] - 二项式等价如果，对于每个长度为 [Formula: see text] 的词 [Formula: see text]，[Formula :see text] 与 [Formula: see text] 和 [Formula: see text] 的子序列（即，作为分散的子词）出现的次数相同。这个概念概括了阿贝尔等价。在本文中，我们研究了由[公式：见正文]-二项式等价性诱导的等价类。我们提供了一种算法来生成单词的 [公式：见正文]-二项式等价类。对于[公式：参见文本]和[公式：参见文本]或更多符号的字母表，由每个[公式：参见文本]-二项式等价类的字典序最少元素组成的语言和单例语言，即， [公式：见文本]-二项式等价类仅限于单个元素，显示为非上下文无关的。作为我们讨论的结果，我们还证明了由 [Formula: see text] 上的自由 nil-[Formula: see text] 群（也称为类 [Formula: see text] 的自由 nilpotent group）的生成器生成的 submonoid ] 生成器同构于自由幺半群 [Formula: see text] 通过 [Formula: see text]-二项式等价的商。