On balanced colorings of the <Emphasis Type="Italic">n</Emphasis>-cube |
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Authors: | William Y C Chen Larry X W Wang |
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Institution: | 1.Center for Combinatorics, LPMC-TJKLC,Nankai University,Tianjin,P.R. China |
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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≥1, the sequence \(\{B_{n,2k}\}_{k=0,1,\ldots,2^{n-1}}\) 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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