### Abstract

Let a_{1}, ..., a_{n} be a sequence of nonzero real numbers with sum zero. A subset B 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 {Mathematical expression} and this bound achieved for n even with a_{j} =(-1)^{j}. Here his conjecture is verified, showing a tight upper bound {Mathematical expression} when n = 2k + 1. The essentially unique extremal configuration is:a_{1} = 2, a_{2} = ... =a_{k} = 1, a_{k+1} = ... =a_{2k+1} = -1.

Original language | English |
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Pages (from-to) | 251-254 |

Number of pages | 4 |

Journal | Graphs and Combinatorics |

Volume | 3 |

Issue number | 1 |

DOIs | |

Publication status | Published - Dec 1 1987 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

Füredi, Z. (1987). The maximum number of balancing sets.

*Graphs and Combinatorics*,*3*(1), 251-254. https://doi.org/10.1007/BF01788547