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A minimum problem for finite sets of real numbers with nonnegative sum. (English) Zbl 1273.11045
Summary: Let \(n\) and \(r\) be two integers such that \(0 < r \leq n\); we denote by \(\gamma(n, r)\) \([\eta(n, r)]\) the minimum [maximum] number of the nonnegative partial sums of a sum \(\sum^n_{1=1} a_i \geq 0\), where \(a_1, \dots, a_n\) are \(n\) real numbers arbitrarily chosen in such a way that \(r\) of them are nonnegative and the remaining \(n - r\) are negative. We study the following two problems:
(P1) which are the values of \(\gamma(n, r)\) and \(\eta(n, r)\) for each \(n\) and \(r\), \(0 < r \leq n\)?
(P2) if \(q\) is an integer such that \(\gamma(n, r) \leq q \leq \eta(n, r)\), can we find \(n\) real numbers \(a_1, \dots, a_n\), such that \(r\) of them are nonnegative and the remaining \(n - r\) are negative with \(\sum^n_{1=1} a_i \geq 0\), such that the number of the nonnegative sums formed from these numbers is exactly \(q\)?

MSC:
11B75 Other combinatorial number theory
05A15 Exact enumeration problems, generating functions
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