Graphs and Combinatorics

, Volume 3, Issue 1, pp 251–254

# The maximum number of balancing sets

• Zoltán Füredi
Article

## Abstract

Leta1, ...,a n be a sequence of nonzero real numbers with sum zero.A subsetB of {1, 2,...,n} is called a balancing set if∑ a b = 0 (b ∈ B). S. Nabeya showed that the number of balancing sets is bounded above by$$\left( {\begin{array}{*{20}c} n \\ {{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-\nulldelimiterspace} 2}} \\ \end{array} } \right)$$ and this bound achieved forn even witha j =(−1) j . Here his conjecture is verified, showing a tight upper bound$$\left( {\begin{array}{*{20}c} {2k} \\ {k - 1} \\ \end{array} } \right)$$ whenn = 2k + 1. The essentially unique extremal configuration is:a1 = 2,a2 = ... =a k = 1,a k+1 = ... =a 2k+1 = -1.

## Keywords

Real Number Nonzero Real Number Extremal Configuration
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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