Summary: For an arithmetic function $f_{0}$, we consider the number $c_{m}(n,k)$ of weighted compositions of $n$ into $k$ parts, where the weights are the values of the $(m-1)^{th}$ invert transform of $f_{0}$. We connect $c_{m}(n,k)$ with $c_{1}(n,k)$ via Pascal matrices. We then relate $c_{m}(n,k)$ to the number of certain restricted words over a finite alphabet. In addition, we develop a method which transfers some properties of restricted words over a finite alphabet to words over a larger alphabet.

05A10, 11B39

binary word, integer composition, restricted word, Pascal matrix