is a three-dimensional array of non-negative integers
denote the number of solid partitions of
Solid partitions and their higher-dimensional generalizations are discussed in the book by Andrews.
[2] Another representation for solid partitions is in the form of Ferrers diagrams.
The Ferrers diagram of a solid partition of
satisfying the condition:[3] For instance, the Ferrers diagram where each column is a node, represents a solid partition of
There is a natural action of the permutation group
on a Ferrers diagram – this corresponds to permuting the four coordinates of all nodes.
This generalises the operation denoted by conjugation on usual partitions.
Given a Ferrers diagram, one constructs the solid partition (as in the main definition) as follows.
that form a solid partition, one obtains the corresponding Ferrers diagram as follows.
nodes given above corresponds to the solid partition with with all other
Define the generating function of solid partitions,
, by The generating functions of integer partitions and plane partitions have simple product formulae, due to Euler and MacMahon, respectively.
However, a guess of MacMahon fails to correctly reproduce the solid partitions of 6.
[3] It appears that there is no simple formula for the generating function of solid partitions; in particular, there cannot be any formula analogous to the product formulas of Euler and MacMahon.
[4] Given the lack of an explicitly known generating function, the enumerations of the numbers of solid partitions for larger integers have been carried out numerically.
There are two algorithms that are used to enumerate solid partitions and their higher-dimensional generalizations.
et al. used an algorithm due to Bratley and McKay.
[5] In 1970, Knuth proposed a different algorithm to enumerate topological sequences that he used to evaluate numbers of solid partitions of all integers
[6] Mustonen and Rajesh extended the enumeration for all integers
[7] In 2010, S. Balakrishnan proposed a parallel version of Knuth's algorithm that has been used to extend the enumeration to all integers
[8] One finds which is a 19 digit number illustrating the difficulty in carrying out such exact enumerations.