Binary Words, N-Color Compositions and Bisection of the Fibonacci Numbers

An n-color composition of n is a composition of n where a part k has k possible colors. It is known that the number of n-color compositions of n is F-2n (the 2nth Fibonacci numbers). Among other objects, F-2n also counts the number of binary words with exactly n - 1 strictly increasing runs and the number of {0, 1, 2} strings of length n - 1 excluding the subword 12. In this note, we show bijections between n-color compositions and these objects. In particular, the bijection between the n-color compositions and the binary words with n - 1 increasing substrings generalizes the classic bijection between compositions and binary words of length n - 1. We also comment on the potential applications of these findings.