Solid Partition

In mathematics, solid partitions are natural generalizations of partitions and plane partitions defined by Percy Alexander MacMahon. A solid partition of $n$ is a three-dimensional array of non-negative integers $n_$ (with indices $i, j, k\geq 1$) such that
:$\sum_ n_=n$
and
:$n_ \leq n_,\quad n_ \leq n_\quad\text\quad n_ \leq n_$ for all $i, j \text k.$
Let $p_3(n)$ denote the number of solid partitions of $n$. 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.

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 $n$ is a collection of $n$ points or ''nodes'', $\lambda=(\mathbf_1,\mathbf_2,\ldots,\mathbf_n)$, with $\mathbf_i\in \mathbb_^4$ satisfying the condition:A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for'' ''m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097–1100.

:Condition FD: If the node $\mathbf=(a_1,a_2,a_3, a_4)\in \lambda$, then so do all the nodes $\mathbf=(y_1,y_2,y_3,y_4)$ with $0\leq y_i\leq a_i$ for all $i=1,2,3,4$.

For instance, the Ferrers diagram
:$\left( \begin 0\\ 0\\ 0 \\ 0 \end \begin 0\\ 0\\ 1 \\ 0 \end \begin 0\\ 1\\ 0 \\ 0 \end \begin1 \\ 0 \\ 0 \\ 0 \end \begin 1 \\ 1\\ 0 \\ 0 \end \right) \ ,$
where each column is a node, represents a solid partition of $5$. There is a natural action of the permutation group $S_4$ on a Ferrers diagram – this corre