## 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 Fm 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 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.

Original language | English |
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Pages (from-to) | 127-140 |

Number of pages | 14 |

Journal | Proceedings of the Indian Academy of Sciences: Mathematical Sciences |

Volume | 124 |

Issue number | 2 |

DOIs | |

State | Published - May 2014 |

## Keywords

- Compositions
- N-color compositions
- Q-generalization

## ASJC Scopus subject areas

- General Mathematics