Abstract
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.
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