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.

KEYWORDS:

• Integer partitions
• polytope of integer partitions
• corner polyhedron
• number of vertices

Acknowledgments

We are deeply obliged to the late Professor V. B. Priezzhev of the Joint Institute for Nuclear Research (JINR), Dubna, Russia, who has recently passed away, for useful discussions. His friendly and encouraging support will be greatly missed. This work could not have been done without computation of vertices performed by A. S. Vroublevski. We thank D. Yang, a graduate student at UCLA at the time, for computing vertices of the master corner polyhedron. We also thank E. S. Zabelova for her assistance in applying Wolfram Mathematica.

Declaration of Interest

No potential conflict of interest was reported by the author.

Note

¹One extra point $t\in P(G_{11},10),$ with $t(5)=1,t(9)=3$ and all other $t(i)=0,$ indicated in [7] as a vertex was excluded in [19].