A solid partition of a nonnegative integer 
is a three-dimensional array
of nonnegative integers, indexed by positive
integers in each coordinate, with finitely many nonzero entries, weakly decreasing
in each coordinate, and having sum 
. The all-zero array
is the unique solid partition of 0. Solid partitions generalize plane
partitions.
The proposed generating function for the number of solid partitions
 |
with exponent 
on 
in the denominator, is incorrect. Its coefficient
of 
is 141, whereas the number of solid partitions of 6 is 140 (Atkin et al. 1967,
pp. 1097-1098). Knuth (1970) extended the tabulation of values, but was unable
to find a correct generating function. The
numbers of solid partitions for 
, 2, ... are 1, 4, 10, 26, 59,
140, ... (OEIS A000293).
-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.