Integer Compositions and Higher-order Conjugation

We consider the classical MacMahon conjugation of compositions or ordered partitions of positive integers. Using both algebraic and graphical methods we provide a natural extension of the standard conjugation of a composition to higher orders. The higher-order conjugates of a composition are obtained by varying the increments used in standard conjugation to turn strings of ones into larger summands and vice versa. It turns out that every nontrivial composition has an integral conjugation order beyond which it is not conjugable. We also discuss recursive conjugation and provide enumeration formulas and combinatorial identities between different classes of compositions.