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?