Divisibility of domino-tiling polynomials by rank-determined linear factors

Prove that for every partition \(\lambda\) of rank \(r\), the domino-tiling generating polynomial \(P_{\lambda}(z;q,t)\) is divisible by \(\prod_{i,j\geq 0,\ i+j<r}(z+q^i t^j)\), with the quotient belonging to \(\mathbb{Z}_{\geq 0}[z,q,t]\).

Background

For a partition $\lambda$, the polynomial $P_{\lambda}(z;q,t)$ is the generating polynomial of domino tilings of the region $R_{\lambda}$, weighted by the statistics $\diags$, $\area$, and $\dinv$. The paper proves an explicit product formula when $\lambda=(n^n)$ is a square partition, yielding factors of the form $z+q^i t^j$.

Computations for all partitions of size at most 9 suggest that the number and indices of these linear factors are governed by the rank of $\lambda$, defined as the side length of its Durfee square. The conjecture predicts the factorization for arbitrary partitions and requires the remaining quotient to have nonnegative integer coefficients. The square-shape case is established by the paper’s product formula, while the general case is left unresolved.