A Method to Count the Positive 3-Subsets in a Set of Real Numbers with Non-Negative Sum

Abstract

be n real numbers with non-negative sum. We show that if n ≥ 12 there exist at least n−1 2 subsets of {a 1 , . . . , a n } with three elements which have non-negative sum.

Sand Piles Models of Signed Partitions with Piles

“…In [22] the authors have introduced and studied a poset ( ( , , related to some extremal combinatorial sums problems (see also [23][24][25][26] for studies on these problems). Such a poset can be seen as a lattice of particular integer partitions with all distinct summands which can be positive or negative, whose maximum positive summand is not exceeding and whose minimum negative summand is not less than −( − .…”

A linear programming approach to the Manickam–Miklós–Singhi conjecture

“…The following sums generated by PropagateNegative (Algorithm 2) must be strictly negative or we have 364 nonnegative sets: 15 x 1 + x 5 + x 9 + x 14 x 1 + x 5 + x 10 + x 13 x 1 + x 6 + x 8 + x 14 x 1 + x 6 + x 9 + x 13 x 1 + x 7 + x 11 + x 12 13 x 2 + x 5 + x 9 + x 12 x 2 + x 6 + x 8 + x 12 x 2 + x 6 + x 10 + x 11 x 2 + x 7 + x 9 + x 11…”

“…The following sums generated by PropagatePositive (Algorithm 3) must be nonnegative or else the associated linear program becomes infeasible: 10 x 3 + x 4 + x 6 + x 11 x 2 + x 4 + x 7 + x 11 x 1 + x 7 + x 9 + x 12 x 1 + x 6 + x 8 + x 13 x 1 + x 5 + x 11 + x 12 x 1 + x 5 + x 8 + x 14 x 1 + x 4 + x 11 + x 13…”

“…The following sums generated by PropagateNegative (Algorithm 2) must be strictly negative or we have at least 286 nonnegative sets: 14 x 1 + x 8 + x 11 + x 14 x 2 + x 4 + x 11 + x 14 x 2 + x 5 + x 9 + x 14 x 2 + x 5 + x 11 + x 13 x 2 + x 6 + x 8 + x 14 x 2 + x 6 + x 9 + x 13 x 2 + x 7 + x 10 + x 12 x 3 + x 4 + x 10 + x 14 x 3 + x 4 + x 12 + x 13 x 3 + x 5 + x 8 + x 14 x 3 + x 5 + x 9 + x 13 x 3 + x 5 + x 10 + x 12 x 3 + x 6 + x 7 + x 14 x 3 + x 6 + x 8 + x 13 x 3 + x 6 + x 9 + x 12 x 3 + x 7 + x 8 + x 12 x 3 + x 7 + x 10 + x 11 x 3 + x 8 + x 9 + x 11 x 4 + x 5 + x 7 + x 14 x 4 + x 5 + x 8 + x 13 x 4 + x 5 + x 9 + x 12 x 4 + x 6 + x 7 + x 13 x 4 + x 6 + x 8 + x 12 x 4 + x 6 + x 9 + x 11 x 5 + x 7 + x 8 + x 11 .…”

“…Conjecture 1 is trivial for k = 1 and is a simple exercise for k = 2. Marino and Chiaselotti [14] proved the conjecture for k = 3. Bier and Manickam [4] showed there exists a minimum integer f (k) such that for all n ≥ f (k), there are at least  n−1 k−1  nonnegative k-sums in any nonnegative sum of n terms.…”