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arXiv · 0910.0903

On Balanced Colorings of the n-Cube

William Y. C. Chen·Larry X. W. Wang
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Abstract

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 $n\geq 1$, the sequence $\{B_{n,2k}\}_{k=0, 1 ... 2^{n-1}}$ 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.

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William Y. C. Chen, Larry X. W. Wang. 2009-10-06. On Balanced Colorings of the n-Cube. https://arxiv.org/abs/0910.0903

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