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 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 F m denotes the Fibonacci number defined by F m =F m−1+F m−2 if m≥2, with F 0=0 and F 1=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 q-generalizations 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.
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MANSOUR, T., SHATTUCK, M. A statistic on n-color compositions and related sequences. Proc Math Sci 124, 127–140 (2014). https://doi.org/10.1007/s12044-014-0166-7
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DOI: https://doi.org/10.1007/s12044-014-0166-7