An identity involving counts of binary matrices

TL;DR

This paper introduces two new identities involving polynomials related to the counts of binary matrices with fixed marginals, offering insights beyond existing approximation methods.

Contribution

It presents novel identities for polynomials connected to the enumeration of binary matrices with given row and column sums, expanding theoretical understanding.

Findings

Derived two identities for polynomials involving $N(p,q)$

Explored consequences of these identities for counting binary matrices

Provided new theoretical tools for analyzing contingency tables

Abstract

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.