125
Views
0
CrossRef citations to date
0
Altmetric
Original Articles

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

Pages 457-466 | Published online: 22 Oct 2021
 

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.

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 tP(G11,10), with t(5)=1,t(9)=3 and all other t(i)=0, indicated in [Citation7] as a vertex was excluded in [Citation19].

Log in via your institution

Log in to Taylor & Francis Online

PDF download + Online access

  • 48 hours access to article PDF & online version
  • Article PDF can be downloaded
  • Article PDF can be printed
USD 61.00 Add to cart

Issue Purchase

  • 30 days online access to complete issue
  • Article PDFs can be downloaded
  • Article PDFs can be printed
USD 360.00 Add to cart

* Local tax will be added as applicable

Related Research

People also read lists articles that other readers of this article have read.

Recommended articles lists articles that we recommend and is powered by our AI driven recommendation engine.

Cited by lists all citing articles based on Crossref citations.
Articles with the Crossref icon will open in a new tab.