# Enumerative geometry and restricted plane partitions

Donaldson-Thomas theory is an enumerative theory for virtual counts of ideal sheaves (with trivial determinant) of the structural sheaf $$\mathcal{O}_{X}$$ of some smooth projective manifold $$X$$.

There is a well-known relation between this theory and the enumerative counting of plane partitions when $$X$$ is a toric Calabi-Yau threefold. The statement for $$X$$ = $$\mathbb{C}^{3}$$ = $$Spec(\mathbb{C}[z_{1},z_{2}$$, $$z_{3}])$$ says that the Donaldson-Thomas theory counts toric-equivariant monomials in $$z_{1},z_{2}$$ and $$z_{3}$$ (as explained in the page 50 of Takagi lectures on Donaldson-Thomas theory). The miraculous fact comes from the fact that there is a bijection between any of the aforementioned monomials and the set of all plane partitions stacked at the origin of $$\mathbb{Z}^{3}$$; to see the bijection we simply identify every box in a given plane partition with a point in $$\mathbb{Z}^{3}$$ and apply the following map $$z_{1}^{a}z_{2}^{b}z_{3}^{c} \mapsto (a,b,c) \in \mathbb{Z}^{3}.$$

Now consider the combinatorics of restricted partitions (plane partitions that are stacked above some rectangle of size $$N\times M$$ where $$N$$ and $$M$$ are not simultaneously infinite as the ones from below) as studied in the page 117 of (1).

This problem has also appeared in gauge theory (see (2) and (3)) in a rather similar way as the counting of ordinary plane partitions has appeared in gauge theory (4).

Question:

What is the algebro-geometric interpretation of the counting of restricted partitions?