  Motivated by work of Gusein-Zade, Luengo, and Melle-Hern\'andez, we study a
specific generating series of arm and leg statistics on partitions, which is
known to compute the Poincar\'e polynomials of Z_3-equivariant Hilbert schemes
of points in the plane, where 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 Z_3 equivariant Hilbert schemes.
