Abstract
Weights of 1 or 0 are assigned to the vertices of the n-cube in n-dimensional Euclidean space. Such an n-cube is called balanced if its center of mass coincides precisely with its geometric center. The seldom-used n-variable form of Pólya's enumeration theorem is applied to express the number N n, 2k of balanced configurations with 2k vertices of weight 1 in terms of certain partitions of 2k. A system of linear equations of Vandermonde type is obtained, from which recurrence relations are derived which are computationally efficient for fixed k. It is shown how the numbers N n, 2k depend on the numbers A n, 2k of specially restricted configurations. A table of values of N n, 2k and A n, 2k is provided for n = 3, 4, 5, and 6. The case in which arbitrary, nonnegative, integral weights are allowed is also treated. Finally, alternative derivations of the main results are developed from the perspective of superposition.
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References
W.K. Clifford, “On the types of compound statement involving four classes,” Mem. Lit. Philos. Soc. Manchester 16 (1877), 88–101.
F. Harary and E.M. Palmer, Graphical Enumeration, Academic Press, New York, 1973.
M.A. Harrison, “The number of transitivity sets of Boolean functions,” SIAM J. Appl. Math. 11 (1963), 806–828.
M.A. Harrison and R.G. High, “On the cycle index of a product of permutation groups,” J. Combin. Theory 4 (1968), 277–299.
W.S. Jevons, The Principles of Science, 2nd ed., Macmillan, London, 1877; reprinted, Dover, New York, 1958.
E.M. Palmer and R.W. Robinson, “Enumeration under two representations of the wreath product,” Acta Math. 131 (1973), 123–143.
E.M. Palmer and R.W. Robinson, “Enumeration of self-dual configurations,” Pacific J. Math. 110 (1984), 203–221.
G. Pólya, “Sur les types des propositions composées,” J Symbolic Logic 5 (1940), 98–103.
R.C. Read, “The enumeration of locally restricted graphs I,” J. London Math. Soc. 34 (1959), 417–436; “The enumeration of locally restricted graphs II,” 35 (1960), 334–351.
J.H. Redfield, “The theory of group-reduced distributions,” Amer. J. Math. 49 (1927), 433–455.
D. Slepian, “On the number of symmetry types of Boolean functions of n variables,” Canad. J. Math. 5 (1953), 185–193.
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Palmer, E., Read, R. & Robinson, R. Balancing the n-Cube: A Census of Colorings. Journal of Algebraic Combinatorics 1, 257–273 (1992). https://doi.org/10.1023/A:1022487918212
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DOI: https://doi.org/10.1023/A:1022487918212