Virtual Counts of Solid Partitions

Abstract

A partition is a finite descending sequence of non negative integers. This concept can be extended to multiple dimensions to form planar and solid partitions. For standard, or line, partitions there is an infinite product formula for the generating function by Euler which gives the number of partitions of a certain size. MacMahon found such a formula for planar partitions. However his conjecture for such an infinite product formula for solid partitions was incorrect for size 6 and up. In this thesis I will look at a conjecture made in an article by Y. Cao and M. Kool, giving a weighted generating function for solid partitions and verify it in many new cases.