Polytopes of partitions of numbers

We study the vertices and facets of the polytopes of partitions of numbers. The partition polytope P n is the convex hull of the set of incidence vectors of all partitions n = x 1 + 2 x 2 + ⋯ + n x n . We show that the sequence P 1 , P 2 , … , P n , … can be treated as an embedded chain. The dynamics of behavior of the vertices of P n , as n increases, is established. Some sufficient and some necessary conditions for a point of P n to be its vertex are proved. Representation of the partition polytope as a polytope on a partial algebra—which is a generalization of the group polyhedron in the group theoretic approach to the integer linear programming—allows us to prove subadditive characterization of the nontrivial facets of P n . These facets ∑ i = 1 n p i x i ≥ p 0 correspond to extreme rays of the cone of subadditive functions p : { 1 , 2 , … , n } → R with additional requirements p 0 = p n and p i + p n − i = p n , 1 ≤ i < n . The trivial facets are explicitly indicated. We also show how all vertices and facets of the polytopes of constrained partitions—in which some numbers are forbidden to participate—can be obtained from those of the polytope P n . All vertices and facets of P n for n ≤ 8 and n ≤ 6 , respectively, are presented.