On Some Numbers Related to Extremal Combinatorial Sum Problems

Abstract

Letn, d,andrbe three integers such that1≤r, d≤n. Chiaselotti (2002) definedγn,d,ras the minimum number of the nonnegative partial sums withdsummands of a sum∑1=1nai≥0, wherea1,…,anarenreal numbers arbitrarily chosen in such a way thatrof them are nonnegative and the remainingn-rare negative. Chiaselotti (2002) and Chiaselotti et al. (2008) determine the values ofγn,d,rfor particular infinite ranges of the integer parametersn, d,andr. 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 ofn, d,andrsuch that(d-1)/dn-1≤r≤(d-1)/dn; (ii)γd+2,d,d=d+1.

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