On balanced colorings of the n-cube

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 B (n,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 na parts per thousand yen1, the sequence is symmetric and unimodal. We give a proof of this conjecture. We also propose a conjecture on the log-concavity of B (n,2k) for fixed k, and by probabilistic method we show that it holds when n is sufficiently large.