Polytopes of partitions of numbers

We study the vertices and facets of the polytopes of partitions of numbers. The partition polytope n P is the convex hull of the set of incidence vectors of all partitions 1 2 2 ... n n x x nx = + + +. We show that the sequence 1 2 n P P P can be treated as an embedded chain. Dynamics of behavior of the vertices of n P , as n increases, is established. Some sufficient and some necessary conditions for a point of n P 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 to prove subadditive characterization of the non-trivial facets of n P. These facets 0 1 n i i i p x p = ≥ ∑ correspond to extreme rays of the cone of subadditive functions :{1, 2,..., } p n → \ with additional requirements 0 n p p = and i n i n p p p − + = , 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 n