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

Authors

Hua Wang, Georgia Southern UniversityFollow
Alex Collins, Georgia State UniversityFollow
Charles Dedrickson, Georgia Southern UniversityFollow

Document Type

Article

Publication Date

5-2013

Publication Title

Fibonacci Quarterly

ISSN

0015-0517

Abstract

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 compositionsof 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 − 1excluding 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.

Recommended Citation

Wang, Hua, Alex Collins, Charles Dedrickson. 2013. "Binary Words, N-Color Compositions and Bisection of the Fibonacci Numbers." Fibonacci Quarterly, 51 (2): 130-136.
https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/214