Number of Vertices of the Polytope of Integer Partitions and Factorization of the Partitioned Number

TL;DR

This paper investigates the number of vertices of the polytope of integer partitions, revealing patterns related to prime and divisible numbers, and proposes a conjecture linking these vertices to divisors of n.

Contribution

It introduces a classification of integers to explain the vertex count patterns and conjectures a divisor-based dependency for the polytope's vertices.

Findings

Vertex count v(n) has a saw-toothed pattern with peaks at prime n.

Divisibility by 3 reduces v(n) due to partitions as convex combinations of three others.

The graph of v(n) stratifies into layers based on integer classes.

Abstract

The polytope of integer partitions of $n$ is the convex hull of the corresponding $n$-dimensional integer points. Its vertices are of importance because every partition is their convex combination. Computation shows intriguing features of $v(n),$ the number of the polytope vertices: its graph has a saw-toothed shape with the highest peaks at prime $n$'s. We explain the shape of $v(n)$ by the large number of partitions of even $n$'s that were counted by N. Metropolis and P. R. Stein. These partitions are convex combinations of two others. We reveal that divisibility of $n$ by 3 also reduces the value of $v(n),$ which is caused by partitions that are convex combinations of three but not two others, and characterize convex representations of such integer points in arbitrary integral polytope. To approach the prime $n$ phenomenon, we use a specific classification of integers and demonstrate that the graph of $v(n)$ is stratified to layers corresponding to resulting classes. Our main conjecture claims that $v(n)$ depends on collections of divisors of $n.$ We also offer an initial argument for that the number of vertices of the Gomory's master corner polyhedron on the cyclic group has features similar to those of $v(n).$