On Balanced Colorings of the n-Cube

الأصل · EN

A 2-coloring of the n-cube in the n-dimensional Euclidean space can be considered as an assignment of weights of 1 or 0 to the vertices. Such a colored n-cube is said to be balanced if its center of mass coincides with its geometric center. Let Bₙ,₂ₖ be the number of balanced 2-colorings of the n-cube with 2k vertices having weight 1. Palmer, Read and Robinson conjectured that for n≥ 1, the sequence {Bₙ,₂ₖ}ₖ₌₀, ₁... ₂ⁿ⁻¹ is symmetric and unimodal. We give a proof of this conjecture. We also propose a conjecture on the log-concavity of Bₙ,₂ₖ for fixed k, and by probabilistic method we show that it holds when n is sufficiently large.