Solid Partition

Solid partitions are generalizations of Plane Partitions. MacMohan (1960) conjectured the Generating Function for the number of solid partitions was

$$f(z)={1\over(1-z)(1-z^2)^3(1-z^3)^6(1-z^4)^{10}\cdots},$$

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, ... (Sloane's A000293).

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, 1912b.

MacMahon, P. A. Combinatory Analysis, Vol. 2. New York: Chelsea, pp. 75-176, 1960.

Sloane, N. J. A. Sequence A000293/M3392 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.