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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 AMS Classification: 05A20, 05D40 Keywords: unimodalily, n-cube, balanced coloring Download: pdf |
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.