Title: Some Formulas for Numbers of Restricted Words,
Abstract: We define a quantity $c_m(n,k)$ as a generalization of the notion of the
composition of the positive integer $n$ into $k$ parts. We proceed to derive
some known properties of this quantity. In particular, we relate two partial
Bell polynomials, in which the sequence of the variables of one polynomial is
the invert transform of the sequence of the variables of the other. We connect
the quantities $c_m(n,k)$ and $c_{m-1}(n,k)$ via Pascal matrices. We then
relate $c_m(n,k)$ with the numbers of some restricted words over a finite
alphabet. We develop a method which transfers some properties of restricted
words over an alphabet of $N$ letters to the restricted words over an alphabet
of $N+1$ letters. Several examples illustrate our findings. Note that all our
results depend solely on the initial arithmetic function $f_0$.