An identity involving counts of binary matrices

In this paper, we prove the proposition shown below and explore some of its consequences. For a sequence p=(p₁,p₂,) of non-negative integers, set |p| = ∑i≥ 1 pᵢ and let [p] = (#: pⱼ=i\)i≥ 1 count the number of occurrences of i≥ 1 in p, so that $|[p]|$ is the number of non-zero elements in p. For p∈ N₀^∞ and x∈ R, we write $p!$ and x^p for the product of factorials ∏i≥ 1pᵢ! and of falling factorials ∏i≥ 1 x^pᵢ, respectively. If p is an integer partition of m, i.e., if $|p|=m$ and p₁≥ p₂ ≥, we write p m. Given two integer partitions p and q of m, let $N(p,q)$ denote the number of |[p]|× |[q]| binary matrices whose row-and column-sums equal p and q, respectively (for $m=0$, this number is to be interpreted as one). Proposition: For q m and x∈ R, we have ∑p m x^|[p]|[p]! N(p,q) = x^qq!.