TITLE: Reference for combinatorics of cell decomposition of the Hilbert scheme of points in the plane
QUESTION [13 upvotes]: It is known from either Morse theory or Bialynicki-Birula decomposition that the fixed points of a ${\mathbb{C}}^*$ action on a smooth algebraic variety over $\mathbb{C}$ determine a cell decomposition of this variety (cells correspond to fixed points by sending each point to a fixed point by the ${\mathbb{C}}^*$ action). Such cell decomposition of the Hilbert scheme of points on the plane $({\mathbb{C}}^2)^{[n]}$ is described in several sources, notably the original papers by G. Ellingsrud and S.A. Stromme and Chapter 5 of the book about Hilbert schemes of points on surfaces by H.Nakajima.
It follows that the fixed points of the torus action (and hence the cells) on $({\mathbb{C}}^2)^{[n]}$ are indexed by partitions of $n$. However, it is hard for me to infer from the above references what is the combinatorial order of these cells (i.e. which cells lie on the boundary of which other cells) -- this should induce a certain order on the partitions of $n$. Is there a reference which would state what this order is explicitly?

REPLY [9 votes]: It might be useful to see quickly why Jeremy's answer, although a very reasonable guess, is wrong. Consider the two partitions $(2,2)$ and $(3,1)$. In refinement order, neither one is greater than the other.
Consider the subscheme of $\mathbb{P}^1 \times \mathbb{C}^2$ cut out by 
$$x^2=u xy + v y^2=y^3=0$$
where $(x,y)$ are the coordinates on $\mathbb{C}^2$ and $(u: v)$ are the homogenous coordinates on $\mathbb{P}^1$. Every fiber over $\mathbb{P}^1$ has length $4$, so this is a flat family, and we get a map $\mathbb{P}^1 \to \mathrm{Hilb}_4(\mathbb{C}^2)$. 
The image of this map is a torus invariant curve in the Hilbert scheme. Its two torus fixed points are the monomial ideals $\langle x^2, y^2 \rangle$ and $\langle x^2, xy, y^3 \rangle$, corresponding to the partitions $(2,2)$ and $(3,1)$.
So, in any Bialynicki-Birula decomposition, one of these partitions must dominate the other.