In mathematics, solid partitions are natural generalizations of partitions and plane partitions defined by Percy Alexander MacMahon.^[1] A solid partition of ${\displaystyle n}$ is a three-dimensional array of non-negative integers ${\displaystyle n_{i,j,k}}$ (with indices ${\displaystyle i,j,k\geq 1}$) such that

${\displaystyle \sum _{i,j,k}n_{i,j,k}=n}$

and

${\displaystyle n_{i+1,j,k}\leq n_{i,j,k},\quad n_{i,j+1,k}\leq n_{i,j,k}\quad {\text{and}}\quad n_{i,j,k+1}\leq n_{i,j,k}}$ for all ${\displaystyle i,j{\text{ and }}k.}$

Let ${\displaystyle p_{3}(n)}$ denote the number of solid partitions of ${\displaystyle n}$. As the definition of solid partitions involves three-dimensional arrays of numbers, they are also called three-dimensional partitions in notation where plane partitions are two-dimensional partitions and partitions are one-dimensional partitions. Solid partitions and their higher-dimensional generalizations are discussed in the book by Andrews.^[2]

Ferrers diagrams for solid partitions

Another representation for solid partitions is in the form of Ferrers diagrams. The Ferrers diagram of a solid partition of ${\displaystyle n}$ is a collection of ${\displaystyle n}$ points or nodes, ${\displaystyle \lambda =(\mathbf {y} _{1},\mathbf {y} _{2},\ldots ,\mathbf {y} _{n})}$, with ${\displaystyle \mathbf {y} _{i}\in \mathbb {Z} _{\geq 0}^{4}}$ satisfying the condition:^[3]

Condition FD: If the node ${\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3},a_{4})\in \lambda }$, then so do all the nodes ${\displaystyle \mathbf {y} =(y_{1},y_{2},y_{3},y_{4})}$ with ${\displaystyle 0\leq y_{i}\leq a_{i}}$ for all ${\displaystyle i=1,2,3,4}$.

For instance, the Ferrers diagram

${\displaystyle \left({\begin{smallmatrix}0\\0\\0\\0\end{smallmatrix}}{\begin{smallmatrix}0\\0\\1\\0\end{smallmatrix}}{\begin{smallmatrix}0\\1\\0\\0\end{smallmatrix}}{\begin{smallmatrix}1\\0\\0\\0\end{smallmatrix}}{\begin{smallmatrix}1\\1\\0\\0\end{smallmatrix}}\right)\ ,}$

where each column is a node, represents a solid partition of ${\displaystyle 5}$. There is a natural action of the permutation group ${\displaystyle S_{4}}$ on a Ferrers diagram – this corresponds to permuting the four coordinates of all nodes. This generalises the operation denoted by conjugation on usual partitions.

Generating function

Let ${\displaystyle p_{3}(0)\equiv 1}$. Define the generating function of solid partitions, ${\displaystyle P_{3}(q)}$, by

${\displaystyle P_{3}(q):=\sum _{n=0}^{\infty }p_{3}(n)q^{n}=1+q+4q^{2}+10q^{3}+26q^{4}+59q^{5}+140q^{6}+\cdots .}$

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]

Exact enumeration using computers

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. The work of Atkin. 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 ${\displaystyle n\leq 28}$.^[6] Mustonen and Rajesh extended the enumeration for all integers ${\displaystyle n\leq 50}$.^[7] In 2010, S. Balakrishnan proposed a parallel version of Knuth's algorithm that has been used to extend the enumeration to all integers ${\displaystyle n\leq 72}$.^[8] One finds

${\displaystyle p_{3}(72)=3464274974065172792\ ,}$

which is a 19 digit number illustrating the difficulty in carrying out such exact enumerations.

Asymptotic behavior

It is conjectured that there exists a constant ${\displaystyle c}$ such that^[9][7][10]

$${\displaystyle \lim _{n\rightarrow \infty }{\frac {\log p_{3}(n)}{n^{3/4}}}=c.}$$

References

External links

• OEIS sequence A000293 (Solid (i.e., three-dimensional) partitions)
• The Solid Partitions Project of IIT Madras
• The Mathworld entry for Solid Partitions