Conjugation of Colored Compositions

An $n$-color composition is a colored composition in which a part of size $m$ may come in $m$ colors. This paper gives a new set of $n$-color-type compositions that admits exhaustive conjugation of its members. Previous attempts at conjugation of $n$-color compositions have yielded partial results at best. Instead of importing the coloring scheme previously used for partitions, we apply colors directly to the parts of compositions while treating any maximal string of 1's as a single part under color 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.