Identities for Partitions of N with Parts from A Finite Set

We show for a prime power number of parts m that the first differences of partitions into at most m parts can be expressed as a non-negative linear combination of partitions into at most m – 1 parts. To show this relationship, we combine a quasipolynomial construction of p(n,m) with a new partition identity for a finite number of parts. We prove these results by providing combinatorial interpretations of the quasipolynomial of p(n,m) and the new partition identity. We extend these results by establishing conditions for when partitions of n with parts coming from a finite set A can be expressed as a non-negative linear combination of partitions with parts coming from a finite set B. We extend this work into Gaussian Polynomials and show that the techniques used can prove asymptotic formulas of partitions with parts from a finite set A