The theoretical part of REU17 project
Table of Contents
1 Combinatorics of partitions
We are interested in study of partitions in dimensions two and three. The main objective is find a combinatorical explanation for coincidence of PT and DT topological vertices. The main tool for us will be Python and Sage library. On the practical level, we plan to create a Python library that would handle 3D partions.
1.1 Partitions
The major treatise on the combinatorics in general and partitions in particular is the book by Stanley: table of contents of volume 1 table of contents of volume 2. Our focus will be the chapter 7 of volume 2. Our goal
1.2 RSK algirithm
RSK algorithm a backbone of enumerative combinatorics and many enumerative formulas for 3D partitions rely on this algorithm. Please read section 11 of chapter 7 of Stanley's book.
1.3 McMahon formula
McMahon formula is the following statement
\begin{equation} \sum_{\pi\in 3D \mbox{ partitions}} q^{|\pi|}=\prod_{k>0}(1-q^k)^{-k}. \end{equation}There are many proofs of this statement. We will look at the proof from Stanley's book, read sections 12, 13 of Chaper 7 of Stanley's book.
1.4 The main problem
There is a natural generalization of the McMahon formula. Instead of usual 3D partitions we can consider the infinite partitions with the infinite ends of given shape. We denote such set by \(3D_\mu\) where \(\mu\) is shape at infinity. It turns out that we can count these partitions as well as usual partitions:
\[ \sum_{\pi\in 3D_\mu} q^{|\pi|}=s_{\mu}(q^\rho)\prod_{k>0}(1-q^k)^{-k}. \]