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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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##### References:
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