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Let S := the nonnegative integer solutions to {$a_1 + ... + a_n = n$}, and center := (1,1,1,...,1). Call a vector v generic if v.s = <=> s = center. Then each generic v defines a positive system in S, the subset { s in S : v.s > }.

Already at n=3 it is possible for one positive system to contain another.

Up to permutation, we may as well take v strictly decreasing. Having done so, is there a reasonable way to classify the maximal positive systems?

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To make the obvious comment, $v$ and $w$ define the same positive system if they lie on the same side of $(s-c)^{\perp}$, where $s$ ranges through the noncentral points of the simplex and $c$ is the center. So your question is whether we can characterize the regions in the complement of this hyperplane arrangement.

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If you replace S by the lattice points in the positive orthant, and forget the condition that the hyperplane passes through a particular point, and require that the hyperplane only cuts off finitely many points, then you are looking at things called "corner cuts" by Onn and Sturmfels in Cutting corners.

Anyway, I'm not sure if this will help, but maybe it'll suggest something.

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