Plane partition

In mathematics and especially in combinatorics, a plane partition is a two-dimensional array of nonnegative integers ${\displaystyle \pi _{i,j}}$ (with positive integer indices i and j) that is nonincreasing in both indices. This means that

${\displaystyle \pi _{i,j}\geq \pi _{i,j+1}}$ and ${\displaystyle \pi _{i,j}\geq \pi _{i+1,j}}$ for all i and j.

Moreover, only finitely many of the ${\displaystyle \pi _{i,j}}$ may be nonzero. Plane partitions are a generalization of partitions of an integer.

A plane partition may be represented visually by the placement of a stack of ${\displaystyle \pi _{i,j}}$ unit cubes above the point (i, j) in the plane, giving a three-dimensional solid as shown in the picture. The image has matrix form

${\displaystyle {\begin{matrix}4&4&3&2&1\\4&3&1&1\\3&2&1\\1\end{matrix}}}$

Plane partitions are also often described by the positions of the unit cubes. From this point of view, a plane partition can be defined as a finite subset ${\displaystyle {\mathcal {P}}}$ of positive integer lattice points (i, j, k) in ${\displaystyle \mathbb {N} ^{3}}$, such that if (r, s, t) lies in ${\displaystyle {\mathcal {P}}}$ and if ${\displaystyle (i,j,k)}$ satisfies ${\displaystyle 1\leq i\leq r}$, ${\displaystyle 1\leq j\leq s}$, and ${\displaystyle 1\leq k\leq t}$, then (i, j, k) also lies in ${\displaystyle {\mathcal {P}}}$.

The sum of a plane partition is

${\displaystyle n=\sum _{i,j}\pi _{i,j}.}$

The sum describes the number of cubes of which the plane partition consists. Much interest in plane partitions concerns the enumeration of plane partitions in various classes. The number of plane partitions with sum n is denoted by PL(n). For example, there are six plane partitions with sum 3

${\displaystyle {\begin{matrix}3\end{matrix}}\qquad {\begin{matrix}2&1\end{matrix}}\qquad {\begin{matrix}1&1&1\end{matrix}}\qquad {\begin{matrix}2\\1\end{matrix}}\qquad {\begin{matrix}1&1\\1\end{matrix}}\qquad {\begin{matrix}1\\1\\1\end{matrix}}}$

so PL(3) = 6.

Plane partitions may be classified by how symmetric they are. Many symmetric classes of plane partitions are enumerated by simple product formulas.