Alexei Oblomkov · June 2017
We are interested in the study of partitions in dimensions two and three. The main objective is to find a combinatorial explanation for the coincidence of the PT and DT topological vertices. The main tool for us will be Python and the Sage library. On the practical level, we plan to create a Python library that handles 3D partitions.
The major treatise on combinatorics in general, and partitions in particular, is Stanley's book: table of contents of volume 1, table of contents of volume 2. Our focus will be chapter 7 of volume 2.
The RSK algorithm is a backbone of enumerative combinatorics, and many enumerative formulas for 3D partitions rely on it. Please read section 11 of chapter 7 of Stanley's book.
The MacMahon formula is the following statement:
\[ \sum_{\pi \in 3D \text{ partitions}} q^{|\pi|} = \prod_{k>0}(1-q^k)^{-k}. \]
There are many proofs of this statement. We will look at the proof from Stanley's book; read sections 12 and 13 of chapter 7.
There is a natural generalization of the MacMahon formula. Instead of the usual 3D partitions we can consider infinite partitions with infinite ends of a given shape. We denote such a set by \(3D_\mu\), where \(\mu\) is the shape at infinity. It turns out that we can count these partitions as well as the usual ones:
\[ \sum_{\pi \in 3D_\mu} q^{|\pi|} = s_{\mu}(q^\rho) \prod_{k>0}(1-q^k)^{-k}. \]
Created 5 June 2017.