In this paper we consider particular generalized compositions of a natural number with a given number of parts. Its number is a weighted polynomial coefficient. The number of all generalized compositions of a natural number is a weighted $r$-generalized Fibonacci number. A relationship between these two numbers will be derived. We shall thus obtain a generalization of the well-known formula connecting Fibonacci numbers with the binomial coefficients.
Generalized Compositions of Natural Numbers
On Number of Compositions of Natural Numbers
Generalized compositions with a fixed number of parts
![QR code for arXiv:1009.3216 [math.CO]](http://oss.arxitics.com/images/favicon.svg)