We use a discrete dynamical model with three evolution rules in order to analyze the structure of a partially ordered set of signed integer partitions whose main properties are actually not known. This model is related to the study of some extremal combinatorial sum problems.
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.