Seminar: Conjugation of Coloured Compositions
Speaker: Augustine Munagi, University of the Witwatersrand, South Africa
Abstract: Compositions of numbers are partitions in which the orders of summands are taken into account, and n-color compositions permit every summand m to appear in m different types or colours. In this talk, we consider a set of n-color-type compositions that admits exhaustive conjugation of its members in the classical tradition of Percy Alexander MacMahon (1854--1929). Previous attempts at conjugation of standard n-color compositions have yielded partial results at best. We condition directly on compositions and devise a set of coloured compositions in which every part dominates its color size while treating every maximal string of 1's as a single part under colour assignment. This leads to the definition of n-color compositions of the second kind. As with ordinary compositions a conjugate may be found by the equivalent techniques of symbolic algebra, zig-zag graph and line graph. We conclude with a derivation of the relevant enumeration formulas.
Keywords: Compositions, Conjugation, Enumeration Formulas