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The Hilbert scheme of $n$ points in the affine plane contains the opensubscheme parametrizing $n$ distinct points in the affine plane, and the closedsubscheme parametrizing ideals of codimension $n$ supported at the origin ofthe affine plane. Both schemes admit Bia{\l}ynicki-Birula decompositions intomoduli spaces of ideals with prescribed lexicographic Gr\"obner deformations.We show that both decompositions are stratifications in the sense that theclosure of each stratum is a union of certain other strata. We show that thecorresponding two partial orderings on the set of of monomial ideals are dualto each other.

Combinatorial duality of Hilbert schemes of points in the affine plane