We consider certain classes of compositions of numbers based on the recently introduced extension of conjugation to higher orders. We use generating functions and combinatorial identities to provide enumeration results for compositions possessing conjugates of a given order. Working under some popular themes in the theory, we show that results for these compositions specialize to standard results in a natural way. We also give a generalization of MacMahon’s identities for inverse-conjugate compositions and discuss inverse-reciprocal compositions.

中文翻译：

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具有高阶共轭的构图组合类

我们基于最近引入的将共轭扩展到高阶来考虑数字的某些类别。我们使用生成函数和组合标识为具有给定顺序的共轭的组合物提供枚举结果。在该理论的一些流行主题下，我们证明了这些组合物的结果以自然的方式专为标准结果。我们还给出了反共轭组合的MacMahon恒等式的一般化，并讨论了反互逆组合。