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

Abstract

The polytope of integer partitions of n is the convex hull of the corresponding n-dimensional integer points. The graph of v(n), the number of the polytope vertices, has a tooth-shaped form with the highest peaks at primes. We explain its shape by the large number of partitions of even n’s that were counted by N. Metropolis and P. R. Stein. We reveal that divisibility of n by 3 also reduces v(n) and characterize convex representations of integer points in arbitrary integral polytope via three other points. Using a specific classification of integers, we demonstrate that the graph of v(n) is stratified into layers corresponding to resulting classes. Our main conjecture claims that the value of v(n) depends on factorization of n. We also offer an argument for that the number of vertices of the master corner polyhedron on the cyclic group has similar features.