User:J D Bowen/Diamond Partitions

Abstract
We explore several forms of diamond plane partitions, notably, a "hollow" 3 by 3 case, and both a 2 by 3 and a 2 by 4 case. We search for functions of the form $f(q,\lambda_{mn})$ which can be used to construct generating functions for the partitions and chains of these partitions as outlined by Matthias Beck in "Diamond Plane Partitions."

The Hollow 3 by 3 Diamond
The "hollow" 3 by 3 diamond is defined as partitions

$\lambda_1\geq {{\lambda_2\geq \lambda_4\geq\lambda_6} \atop {\lambda_3\geq\lambda_5\geq\lambda_7}} \geq \lambda_8$.

The desired auxillary function for this partition is

$f(q,\lambda_8)=\sum_{\text{relations shown}}q^{\Sigma \lambda_j}$

$=\sum_{\lambda_7\geq\lambda_8} q^{\lambda_7} \sum_{\lambda_6\geq\lambda_8} q^{\lambda_6}\sum_{\lambda_5\geq\lambda_7}q^{\lambda_5}\sum_{\lambda_4\geq\lambda_6}q^{\lambda_4}\sum_{\lambda_3\geq\lambda_5}q^{\lambda_3}\sum_{\lambda_2\geq\lambda_4}q^{\lambda_2}\sum_{\lambda_1\geq\text{max}(\lambda_2,\lambda_3)}q^{\lambda_1} \ $

$=\sum_{\lambda_7\geq\lambda_8} q^{\lambda_7} \sum_{\lambda_6\geq\lambda_8} q^{\lambda_6}\sum_{\lambda_5\geq\lambda_7}q^{\lambda_5}\sum_{\lambda_4\geq\lambda_6}q^{\lambda_4}\sum_{\lambda_3\geq\lambda_5}q^{\lambda_3}\sum_{\lambda_2\geq\lambda_4}q^{\lambda_2} \frac{q^{\text{max}(\lambda_2,\lambda_3)}}{1-q} \ $

$=\sum_{\lambda_7\geq\lambda_8} q^{\lambda_7} \sum_{\lambda_6\geq\lambda_8} q^{\lambda_6}\sum_{\lambda_5\geq\lambda_7}q^{\lambda_5}\sum_{\lambda_4\geq\lambda_6}q^{\lambda_4}\sum_{\lambda_3\geq\lambda_5}q^{\lambda_3}

\left({ q^{\lambda_3}\frac{q^{\lambda_4}-q^{\lambda_3}}{1-q}+\frac{q^{2\lambda_3}}{1-q^2}}\right) \ $