The maximum number of balancing sets

Abstract

Leta ₁, ...,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:a ₁ = 2,a ₂ = ... =a _k = 1,a _k+1 = ... =a _2k+1 = -1.