Partitions and compositions over finite fields

In this paper we find exact formulas for the numbers of partitions and compositions of an element into $m$ parts over a finite field, i.e. we find the number of nonzero solutions of the equation $x_1+x_2+...+x_m=z$ over a finite field when the order does not matter and when it does, respectively. We also give an application of our results in the study of polynomials of prescribed ranges over finite fields.