Combinatorial duality of Hilbert schemes of points in the affine plane

The Hilbert scheme of $n$ points in the affine plane contains the open subscheme parametrizing $n$ distinct points in the affine plane, and the closed subscheme parametrizing ideals of codimension $n$ supported at the origin of the affine plane. Both schemes admit Bia{\l}ynicki-Birula decompositions into moduli spaces of ideals with prescribed lexicographic Gr\"obner deformations. We show that both decompositions are stratifications in the sense that the closure of each stratum is a union of certain other strata. We show that the corresponding two partial orderings on the set of of monomial ideals are dual to each other.