A plane partition of a positive integer $n$ is a two-dimensional array $\pi$ of non-negative integers ${\displaystyle n_{i,j}} $ (with positive integer indices $i$ and $j$) that is non-increasing in both indices, i.e. satisfying \[ {\displaystyle n_{i,j}\geq n_{i,j+1}\quad {\mbox{and}}\quad n_{i,j}\geq n_{i+1,j}\,}, \] for all $i$ and $j$, and for which only finitely many of the $n_{i,j}$ are non-zero, and for which \[ {\displaystyle \sum _{i,j}n_{i,j}=n.\,} \] Plane partitions were introduced by Major Percy MacMahon in [MacM1]. Let $pl(n)$ denote the number of plane partitions of $n$. MacMahon conjectured that the generating function was given by \[ \sum_{n=0}^{\infty}pl(n)q^n=\prod_{k=1}^{\infty}\frac{1}{(1-q^k)^k}, \] and this result follows from another result of MacMahon, and was also later proved by Andrews and Paule [AP], using a method ( MacMahon’s omega calculus) which MacMahon had tried to make work, but failed.

A plane partition may be represented graphically by a 3-D arrangement of cubes (or 3-D Young diagram), by replacing the part $n_{i,j}$ in the array by a stack of $n_{i,j}$ cubes. For example, the following plane partition of $n=155$, \begin{equation}\label{ppeq1} \begin{array}{cccccccccccc} 6 & 5 & 4 & 4 & 4 & 3 & 3 & 2 & 2 & 2 & 1 & 1 \\ 5 & 5 & 4 & 4 & 3 & 3 & 2 & 2 & 2 & 1 & 1 & 1 \\ 5 & 4 & 4 & 4 & 3 & 3 & 2 & 2 & 2 & 1 & 1 & \text{} \\ 4 & 4 & 4 & 3 & 3 & 3 & 2 & 2 & \text{} & \text{} & \text{} & \text{} \\ 3 & 3 & 3 & 3 & 2 & 2 & 1 & 1 & \text{} & \text{} & \text{} & \text{} \\ 3 & 2 & 2 & 1 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 2 & 1 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ \end{array} \end{equation} is represented by the following arrangement:

Note: This diagram, and the others below, were created using the Mathematica notebook created by Wouter Meeussen and Eric W. Weisstein, which is available for download at the Wikipedia page [W1].

Just as with regular integer partitions, it is possible to consider various kinds of restricted plane partition function. A survey of some of the results for some of these restricted plane partition functions may be found in the survey paper of Krattenthaler [K], in which 10 symmetry classes of plane partitions are listed, some of which are described below.

A question MacMahon asked himself was the following.
Let $a$, $b$ and $c$ be positive integers, and let $pp_{a,b,c}(n)$ denote the number of plane partitions of $n$ into parts $n_{i,j}$ such that $1\leq i \leq a$, $1\leq j \leq b$, and $1\leq n_{i,j} \leq c$ (alternatively, $pp_{a,b,c}(n)$ is the number of plane partitions of $n$ whose 3-D Young diagram fits inside a cube of size $a\times b \times c$). MacMahon showed that \begin{equation}\label{ppeq2} \sum_{n}pp_{a,b,c}(n)q^n = \prod_{i=1}^a \prod_{j=1}^b \prod_{k=1}^c \frac{1-q^{i+j+k-1}}{1-q^{i+j+k-2}}. \end{equation} MacMahon'sformula at \eqref{ppeq1} follows from \eqref{ppeq2} upon letting $a,b,c\to \infty$.

A plane partition is symmetric if $n_{i,j}=n_{j,i}$, for all $i$ and $j$. Here is one such partition of $n=125$, which is also displayed graphically below: \begin{equation}\label{ppeq3} \begin{array}{cccccc} 6 & 6 & 6 & 5 & 5 & 3 \\ 6 & 5 & 5 & 4 & 4 & 3 \\ 6 & 5 & 5 & 3 & 3 & 3 \\ 5 & 4 & 3 & 2 & 2 & 1 \\ 5 & 4 & 3 & 2 & 1 & \text{} \\ 3 & 3 & 3 & 1 & \text{} & \text{} \\ \end{array} \end{equation}

If $SPP_n$ denotes the number of symmetric partitions whose (3-D) Young diagram fits inside a cube of size $n$, then \[ SPP_n = \prod_{i=1}^n \prod_{j=1}^n \frac{i+j+i-1}{i+j+i-2}. \] If $spp_{n,n,m}(k)$ denotes the number of symmetric plane partitions of $k$ whose Young diagram fits inside an $n\times n \times m$ box, then \begin{equation}\label{ppeq4} \sum_{k}spp_{n,n,m}(k) q^k= \prod_{i=1}^n \frac{1-q^{m+2i-1}}{1-q^{2i-1}} \prod_{1\leq i < j\leq n} \frac{1-q^{2(m+i+j-1)}}{1-q^{2(i+j-1)}}. \end{equation} This formula was conjectured by MacMahon and proved by Andrews [A] and MacDonald [M].

Another class of plane partitions comprise totally symmetric plane partitions (TSPP), which are plane partitions whose Young diagrams remain invariant under all permutations of the axes. One example is the following plane partition of $n=105$ \begin{equation}\label{ppeq5} \begin{array}{cccccc} 6 & 6 & 6 & 5 & 5 & 3 \\ 6 & 5 & 5 & 3 & 3 & 1 \\ 6 & 5 & 4 & 3 & 2 & 1 \\ 5 & 3 & 3 & 1 & 1 & \text{} \\ 5 & 3 & 2 & 1 & 1 & \text{} \\ 3 & 1 & 1 & \text{} & \text{} & \text{} \\ \end{array}, \end{equation} which has Young diagram

If $TSPP_n(k)$ denotes the number of totally symmetric plane partitions of $k$ such that the Young diagram fits inside a cube of side $n$, then the generating function has the form \[ \sum_{k}TSPP_n(k)q^k = \prod_{1\leq i\leq j\leq k \leq n} \frac{1-q^{i+j+k-1}}{1-q^{i+j+k-2}}. \]

The complement of a plane partition with Young diagram inside an $a \times b \times c$ box is the partition represented by the complement of the Young diagram in the $a \times b \times c$ box, suitably re-oriented.
Another class of plane partitions are totally symmetric self-complementary plane partitions (TSSCPP), which are totally symmetric plane partitions that are equal to their own complement. An example is this totally symmetric self-complementary plane partition of $n=108$ \begin{equation}\label{ppeq6} \begin{array}{cccccc} 6 & 6 & 6 & 5 & 5 & 3 \\ 6 & 5 & 5 & 4 & 3 & 1 \\ 6 & 5 & 4 & 3 & 2 & 1 \\ 5 & 4 & 3 & 2 & 1 & \text{} \\ 5 & 3 & 2 & 1 & 1 & \text{} \\ 3 & 1 & 1 & \text{} & \text{} & \text{} \\ \end{array}, \end{equation} with Young diagram

The number of totally symmetric self-complementary plane partitions with Young diagrams that fit inside a $(2n)\times (2n)\times (2n)$ is equal to \[ \prod_{i=0}^{n-1} \frac{(3i+1)!}{(a+i)!}. \]

Another sub-class of totally symmetric plane partitions, which is not one of the ten classes listed in [K], is the class of 1-shell totally symmetric plane partitions, which were examined by Blecher [B]. Such an object is a totally symmetric plane partition has a self- conjugate first row/column (as an ordinary partition) and all other entries are 1. One example is this plane partition of $n=76$ \begin{equation}\label{ppeq7} \begin{array}{cccccc} 6 & 6 & 6 & 5 & 5 & 3 \\ 6 & 1 & 1 & 1 & 1 & 1 \\ 6 & 1 & 1 & 1 & 1 & 1 \\ 5 & 1 & 1 & 1 & 1 & \text{} \\ 5 & 1 & 1 & 1 & 1 & \text{} \\ 3 & 1 & 1 & \text{} & \text{} & \text{} \\ \end{array}, \end{equation} with Young diagram

Amongst other results, Blecher showed in [B] that if $f(n)$ denotes the number of 1-shell totally symmetric plane partitions of the integer $n$, then \begin{equation}\label{ppeq8} \sum_{n=0}^{\infty}f(n)q^n =1+ \sum_{k=1}^{\infty} q^{3k-2}\prod_{i=0}^{k-2}(1+q^{6i+3}). \end{equation}

Just as with various other partition functions, there are congruences satisfied by $f(n)$ for $n$ in various arithmetic progressions, which have been found by several authors. For example, Chern showed in [C] that for all $n \geq 0$, \begin{align}\label{ppeq9} f(1250n + 125) &\equiv 0(\mod 125),\\ f(1250n + 1125) &\equiv 0(\mod 125),\\ f(2750n + 825) &\equiv 0(\mod 11),\\ f(2750n + 1925) &\equiv 0(\mod 11). \end{align}

[A] G. Andrews, Plane partitions (I): the MacMahon conjecture, Studies in Foundations and Combinatorics, Advances in Mathematics Supplementary Studies 1 (1978) 131–150.

[AP] G. E. Andrews and P. Paule, MacMahon’s partition analysis, XII: Plane partitions, J. London Math. Soc. (2) 76 (2007), 647—666.

[B] A. Blecher, Geometry for totally symmetric plane partitions (TSPPs) with self-conjugate main diagonal, Util. Math. 88 (2012), 223–235

[C] S. Chern, Congruences for 1-shell totally symmetric plane partitions, Integers 17 (2017), Paper No. A21, 7 pp.

[K] C. Krattenthaler, Plane partitions in the work of Richard Stanley and his school. The mathematical legacy of Richard P. Stanley, 231–261, Amer. Math. Soc., Providence, RI, 2016.

[M] I.G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edn., Oxford University Press, 1995.

[MacM1] P.A. MacMahon, Memoir on the theory of the partition of numbers, I, Lond. Phil. Trans. (A) 187 (1897), 619–673.

[MacM2] P.A. MacMahon “Partitions of numbers whose graphs possess symmetry,” Trans. Cambridge Phil. Soc. 17 (1898–99) 149–170.

[W1] http://mathworld.wolfram.com/PlanePartition.html.

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