On Some Numbers Related to Extremal Combinatorial Sum Problems

Let n, d, and r be three integers such that 1≤r, d≤n. Chiaselotti (2002) defined γn,d,r as the minimum number of the nonnegative partial sums with d summands of a sum ∑1=1nai≥0, where a1,…,an are n real numbers arbitrarily chosen in such a way that r of them are nonnegative and the remaining n-r are negative. Chiaselotti (2002) and Chiaselotti et al. (2008) determine the values of γn,d,r for particular infinite ranges of the integer parameters n, d, and r. In this paper we continue their approach on this problem and we prove the following results: (i) γ(n,d,r)≤(rd)+(rd-1) for all values of n, d, and r such that (d-1)/dn-1≤r≤(d-1)/dn; (ii) γd+2,d,d=d+1