A statistic on n-color compositions and related sequences

A composition of a positive integer in which a part of size n may be assigned one of n colors is called an n-color composition. Let a_m denote the number of n-color compositions of the integer m. It is known that a_m = F_2m for all m ≥ 1, where Fm denotes the Fibonacci number defined by F_m = F_m-1 + F_m-2 if m ≥ 2, with F₀ = 0 and F₁ = 1. A statistic is studied on the set of n-color compositions of m, thus providing a polynomial generalization of the sequence F_2m. The statistic may be described, equivalently, in terms of statistics on linear tilings and lattice paths. The restriction to the set of n-color compositions having a prescribed number of parts is considered and an explicit formula for the distribution is derived. We also provide qgeneralizations of relations between a_m and the number of self-inverse n-compositions of 2m + 1 or 2m. Finally, we consider a more general recurrence than that satisfied by the numbers a_m and note some particular cases.