On congruence properties of the coefficients of Gaussian polynomials

Abstract:

In this paper we establish infinite families of prime divisibility properties for the partition function $p(n,m,N)$ enumerating partitions into at most $m$ parts with no part larger than $N$. These partition numbers are the coefficients of the Gaussian polynomials ${N+m\brack m}$. The proof of the main theorem connects three fundamental partition functions: the general partition function $p(n)$, the restricted partition function $p(n,m)$ which counts partitions of $n$ into at most $m$ parts, and $p(n,m,N)$.