Integer Partitions from a Geometric Viewpoint

The study of partitions and compositions (i.e., ordered partitions) of integers goes back centuries and has applications in various areas within and outside of mathematics. Partition analysis is full of beautiful – and sometimes surprising – identities, starting with Euler's classic theorem that the number of partitions of an integer k into odd parts equals the number of partitions of k into distinct parts.

Motivated by work of George Andrews, Peter Paule, and coauthors from the last 1 1/2 decades, we will show how one can shed light on certain classes of partition identities by interpreting partitions as integer points in polyhedra. Our approach yields both "short" proofs of known results and new theorems.