Partitions and compositions over finite fields

TL;DR

This paper derives exact formulas for counting partitions and compositions of elements into parts over finite fields, with applications to polynomials of prescribed ranges, advancing combinatorial understanding in finite field contexts.

Contribution

It provides the first explicit formulas for partitions and compositions over finite fields and applies these results to polynomial range problems.

Findings

Exact formulas for partitions over finite fields

Exact formulas for compositions over finite fields

Application to polynomials with prescribed ranges

Abstract

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.