#### Title

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

#### 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