Journal of Integer Sequences, Vol. 21 (2018), Article 18.8.5

Integer Compositions and Higher-Order Conjugation

Augustine O. Munagi
School of Mathematics
University of the Witwatersrand
Wits 2050, Johannesburg
South Africa


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.

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(Concerned with sequences A000045 A000079 A000930 A003269 A003520 A005708 A005709 A005710 A005711 A017898 A017899 A017900 A017901 A017902 A017903 A017904 A027934 A055389 A145018 A233583.)

Received May 29 2018; revised versions received August 30 2018; August 31 2018. Published in Journal of Integer Sequences, November 25 2018.

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