For an arithmetic function f₀, 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 f₀. We connect c_m(n,k) with c₁(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.

Several examples illustrate our findings. Some examples concern binomial coefficients and Fibonacci numbers. Some examples also extend the classical results about weighted compositions of Hoggatt and Lind. In each example, we derive an explicit formula for c_m(n,k).