# “Positive systems” in n * the (n-1)-simplex

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 = v.center <=> s = center. Then each generic v defines a positive system in S, the subset { s in S : v.s > v.center }.

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.