COMPUTING VERTICES OF INTEGER PARTITION POLYTOPES

Abstract

The paper describes a method of generating vertices of the polytopes of integer partitions that was used by the authors to calculate all vertices and support vertices of the partition polytopes for all n ≤ 105 and all knapsack partitions of n ≤ 165. The method avoids generating all partitions of n. The vertices are determined with the help of sufficient and necessary conditions; in the hard cases, the well-known program Polymake is used. Some computational aspects are exposed in more detail. These are the algorithm for checking the criterion that characterizes partitions that are convex combinations of two other partitions; the way of using two combinatorial operations that transform the known vertices to the new ones; and employing the Polymake to recognize a limited number (for small n) of partitions that need three or more other partitions for being convexly expressed. We discuss the computational results on the numbers of vertices and support vertices of the partition polytopes and some appealing problems these results give rise to.