Counting 3-way contingency tables via quiver semi-invariants

Abstract

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 $\mathbf{a}=(a_1,\dots ,a_m)\in {\mathbb{N}}^m$ and $\mathbf{b}=(b_1,\dots ,b_n)\in {\mathbb{N}}^n$. 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 ${\mathcal{Q}}_{m,n}^p$ be the $p$-complete bipartite quiver with $m$ source vertices, $n$ sink vertices, and $p$ arrows from each source to each sink. Let $1$ denote the dimension vector of ${\mathcal{Q}}_{m,n}^p$ that takes value $1$ at every vertex of ${\mathcal{Q}}_{m,n}^p$, 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 ${\mathcal{Q}}_{m,n}^p$. We begin by realizing ${\mathbf{T}}_{\mathbf{a},\mathbf{b}}$ as the dimension of the space of semi-invariants associated to $({\mathcal{Q}}_{m,n}^p,1,{\theta }_{\mathbf{a},\mathbf{b}})$. 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.

Added acknowledgements