A statistic on n-color compositions and related sequences

Abstract

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 am denote the number of n-color compositions of the integer m. It is known that am=F2m for all m≥1, where Fm denotes the Fibonacci number defined by Fm=Fm−1+Fm−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 F2m. 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 q-generalizations of relations between am 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 am and note some particular cases.