Generalized Compositions and Weighted Fibonacci Numbers

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 with a fixed number of parts

We investigate compositions of a positive integer with a fixed number of parts, when there are several types of each natural number. These compositions produce new relationships among binomial coefficients, Catalan numbers, and numbers of the Catalan triangle