A Minimum Problem for Finite Sets of Real Numbers with Nonnegative Sum

Let n and r be two integers such that 0 < r ≤ n; we denote by γ(n, r) [η(n, r)] the minimum [maximum] number of the non-negative partial sums of a sum ∑ n 1=1 ai ≥ 0, where a1, · · · , an are n real numbers arbitrarily chosen in such a way that r of them are non-negative and the remaining n − r are negative. Inspired by some interesting extremal combinatorial sum problems raised by Manickam, Miklös and Singhi in 1987 [12] and 1988 [13] we study the following two problems: (P1) which are the values of γ(n, r) and η(n, r) for each n and r, 0 < r ≤ n? (P2) if q is an integer such that γ(n, r) ≤ q ≤ η(n, r), can we find n real numbers a1, · · · , an, such that r of them are non-negative and the remaining n−r are negative with ∑ n 1=1 ai ≥ 0, such that the number of the non-negative sums formed from these numbers is exactly q? We prove that the solution of the problem (P1) is given by γ(n, r) = 2 and η(n, r) = 2−2 . We provide a partial result of the latter problem showing that the answer is affirmative for the weighted boolean maps. With respect to the problem (P2) such maps (that we will introduce in the present paper) can be considered a generalization of the multisets a1, · · · , an with ∑ n 1=1 ai ≥ 0. More precisely we prove that for each q such that γ(n, r) ≤ q ≤ η(n, r) there exists a weighted boolean map having exactly q positive boolean values.