Topology of $\mathbb{Z}_3$-Equivariant Hilbert Schemes

Motivated by work of Gusein-Zade, Luengo, and Melle-Hernández, we study a specific generating series of arm and leg statistics on partitions, which is known to compute the Poincaré polynomials of $\mathbb{Z}_3$-equivariant Hilbert schemes of points in the plane, where $\mathbb{Z}_3$ acts diagonally. This generating series has a conjectural product formula, a proof of which has remained elusive over the last ten years. We introduce a new combinatorial correspondence between partitions of $n$ and $\{1,2\}$-compositions of $n$, which behaves well with respect to the statistic in question. As an application, we use this correspondence to compute the highest Betti numbers of the $\mathbb{Z}_3$-equivariant Hilbert schemes.