An identity involving counts of binary matrices

In the context of generating uniform random contingency tables with pre-specified marginals, the number of (binary) matrices with given row- and column-sums is a well-studied object in the literature. We will denote this number by $N(p,q)$, where $p$ and $q$ are the vectors of row- and column-sums. The existing literature is mainly focused on computing or approximating $N(p,q)$. In this paper, we present two identities for polynomials whose coefficients depend on the $N(p,q)$ and explore some consequences.