Linking partition combinatorics to the geometry of Hilbert schemes

One of the key objects in studying the Hilbert Scheme of points in the plane is a torus action of $(\mathbb{C}^*)^2$. The fixed points of this action correspond to monomial ideals in $\mathbb{C}[x,y]$, and this gives a connection between the geometry of Hilbert schemes and partition combinatorics. Using this connection, one can extract identities in partition combinatorics from algebro-geometric information and vice versa. I will give some examples of combinatorial identities where as yet the only proofs we have rely on the geometry of Hilbert schemes. If there is time, I will also sketch out a hope that such identities can also be seen by representations of appropriately chosen algebras.