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Solid Partition

Solid partitions are generalizations of plane partitions. MacMahon (1960) conjectured the generating function for the number of solid partitions was

f(z)=1/((1-z)(1-z^2)^3(1-z^3)^6(1-z^4)^(10)...),

but this was subsequently shown to disagree at n=6 (Atkin et al. 1967). Knuth (1970) extended the tabulation of values, but was unable to find a correct generating function. The first few values are 1, 4, 10, 26, 59, 140, ... (OEIS A000293).

See also

Partition Function P

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References

Atkin, A. O. L.; Bratley, P.; Macdonald, I. G.; and McKay, J. K. S. "Some Computations for m-Dimensional Partitions." Proc. Cambridge Philos. Soc. 63, 1097-1100, 1967.Knuth, D. E. "A Note on Solid Partitions." Math. Comput. 24, 955-961, 1970.MacMahon, P. A. "Memoir on the Theory of the Partitions of Numbers. VI: Partitions in Two-Dimensional Space, to which is Added an Adumbration of the Theory of Partitions in Three-Dimensional Space." Phil. Trans. Roy. Soc. London Ser. A 211, 345-373, 1912.MacMahon, P. A. Combinatory Analysis, Vol. 2. New York: Chelsea, pp. 75-176, 1960.Sloane, N. J. A. Sequence A000293/M3392 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Solid Partition

Cite this as:

Weisstein, Eric W. "Solid Partition." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SolidPartition.html

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