Counting 3-way contingency tables via quiver semi-invariants

Let $\mathbf{T}_{\mathbf{a},\mathbf{b}}$ be the number of $3$-way contingency tables of size $m \times n \times p$ with two of its three plane-sum margins fixed by $\mathbf{a}=(a_1, \ldots, a_m) \in \mathbb{N}^m$ and $\mathbf{b}=(b_1, \ldots, b_n) \in \mathbb{N}^n$. When $p=1$, this is the number of $m \times n$ non-negative integer matrices whose row and column sums are fixed by $\mathbf{a}$ and $\mathbf{b}$. In this paper, we study the numbers $\mathbf{T}_{\mathbf{a},\mathbf{b}}$ through the lens of quiver invariant theory. Let $\mathcal{Q}^{p}_{m,n}$ be the $p$-complete bipartite quiver with $m$ source vertices, $n$ sink vertices, and $p$ arrows from each source to each sink. Let $\mathbf{1}$ denote the dimension vector of $\mathcal{Q}^{p}_{m,n}$ that takes value $1$ at every vertex of $\mathcal{Q}^{p}_{m,n}$, and let $\theta_{\mathbf{a}, \mathbf{b}}$ denote the integral weight that assigns $a_i$ to the $i^{th}$ source vertex and $-b_j$ to the $j^{th}$ sink vertex of $\mathcal{Q}^{p}_{m,n}$. We begin by realizing $\mathbf{T}_{\mathbf{a},\mathbf{b}}$ as the dimension of the space of semi-invariants associated to $(\mathcal{Q}^{p}_{m,n}, \mathbf{1}, \theta_{\mathbf{a}, \mathbf{b}})$. Using this connection and methods from quiver invariant theory, we show that $\mathbf{T}_{\mathbf{a},\mathbf{b}}$ is a parabolic Kostka coefficient. In the case $p=1$, this recovers the formula for the number of the $m \times n$ contingency tables with row and column sums fixed by $\mathbf{a}$ and $\mathbf{b}$, which in the classical $2$-way setting can also be obtained via the Robinson-Schensted-Knuth correspondence.