Goulden–Jackson's Matching–Jack conjecture

Let $c^\rho_{\tau, u}(\alpha)$ be the coefficients defined by

$$\nabla(\boldsymbol{p},\boldsymbol{q})p_\rho=\sum_{\tau,\nu\vdash n}c^\rho_{\tau,\nu}(\alpha)p_\tau q_\nu.$$

Here $b:=\alpha-1$ is the shifted Jack parameter. Goulden–Jackson's conjecture. The coefficients satisfy $c^\rho_{\tau,\nu}(\alpha)\in\mathbb N[b]$. This is the Matching–Jack conjecture, a long-standing positivity problem in Jack theory. It remains open despite many partial results.