On Some Numbers Related to Extremal Combinatorial Sum Problems. (3rd March 2014)

• Record Type: Journal Article
• Authors: Petrassi, D.
• Other Names: Petit Franck Academic Editor.
• Is Part Of: Journal of discrete mathematics. Volume 2014(2014)
• Publication Date: 2014-03-03
• Subjects: Computer science -- Mathematics -- Periodicals; Computer science -- Mathematics; Periodicals; 511.1
• DOI: 10.1155/2014/979171
• Languages: English
• ISSNs: 2090-9837

Abstract

Abstract : Let n, d, and r be three integers such that1 ≤ r, d ≤ n . Chiaselotti (2002) definedγ n, d, r as the minimum number of the nonnegative partial sums with d summands of a sum∑ 1 = 1 n a i ≥ 0, wherea 1, …, a n are n real numbers arbitrarily chosen in such a way that r of them are nonnegative and the remainingn - r are negative. Chiaselotti (2002) and Chiaselotti et al. (2008) determine the values ofγ n, 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 ) ≤ ( r d ) + ( r d - 1 ) for all values of n, d, and r such that(d - 1) / d n - 1 ≤ r ≤ (d - 1) / d n ; (ii)γ d + 2, d, d = d + 1 .