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The polytope of integer partitions of $n$ is the convex hull of thecorresponding $n$-dimensional integer points. Its vertices are of importancebecause every partition is their convex combination. Computation showsintriguing features of $v(n),$ the number of the polytope vertices: its graphhas a saw-toothed shape with the highest peaks at prime $n$'s. We explain theshape of $v(n)$ by the large number of partitions of even $n$'s that werecounted by N. Metropolis and P. R. Stein. These partitions are convexcombinations of two others. We reveal that divisibility of $n$ by 3 alsoreduces the value of $v(n),$ which is caused by partitions that are convexcombinations of three but not two others, and characterize convexrepresentations of such integer points in arbitrary integral polytope. Toapproach the prime $n$ phenomenon, we use a specific classification of integersand demonstrate that the graph of $v(n)$ is stratified to layers correspondingto resulting classes. Our main conjecture claims that $v(n)$ depends oncollections of divisors of $n.$ We also offer an initial argument for that thenumber of vertices of the Gomory's master corner polyhedron on the cyclic grouphas features similar to those of $v(n).$

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