§26.12 Plane Partitions

A plane partition, $\pi$, of a positive integer $n$, is a partition of $n$ in which the parts have been arranged in a 2-dimensional array that is weakly decreasing (nonincreasing) across rows and down columns. Different configurations are counted as different plane partitions. As an example, there are six plane partitions of 3:

An equivalent definition is that a plane partition is a finite subset of $\mathbb{N}\times \mathbb{N}\times \mathbb{N}$ with the property that if $(r,s,t)\in \pi$ and $(1,1,1)\le (h,j,k)\le (r,s,t)$, then $(h,j,k)$ must be an element of $\pi$. Here $(h,j,k)\le (r,s,t)$ means $h\le r$, $j\le s$, and $k\le t$. It is useful to be able to visualize a plane partition as a pile of blocks, one block at each lattice point $(h,j,k)\in \pi$. For example, Figure 26.12.1 depicts the pile of blocks that represents the plane partition of 75 given by (26.12.2).

The number of plane partitions of $n$ is denoted by $\mathrm{pp}\left(n\right)$, with $\mathrm{pp}\left(0\right)=1$. See Table 26.12.1.

We define the $r\times s\times t$ box $B(r,s,t)$ as

Then the number of plane partitions in $B(r,s,t)$ is

A plane partition is symmetric if $(h,j,k)\in \pi$ implies that $(j,h,k)\in \pi$. The number of symmetric plane partitions in $B(r,r,t)$ is

A plane partition is cyclically symmetric if $(h,j,k)\in \pi$ implies $(j,k,h)\in \pi$. The plane partition in Figure 26.12.1 is an example of a cyclically symmetric plane partition. The number of cyclically symmetric plane partitions in $B(r,r,r)$ is

or equivalently,

A plane partition is totally symmetric if it is both symmetric and cyclically symmetric. The number of totally symmetric plane partitions in $B(r,r,r)$ is

The complement of $\pi \subseteq B(r,s,t)$ is ${\pi }^c=\{(h,j,k)|(r-h+1,s-j+1,t-k+1)\notin \pi \}$. A plane partition is self-complementary if it is equal to its complement. The number of self-complementary plane partitions in $B(2r,2s,2t)$ is

in $B(2r+1,2s,2t)$ it is

in $B(2r+1,2s+1,2t)$ it is

A plane partition is transpose complement if it is equal to the reflection through the $(x,y)$-plane of its complement. The number of transpose complement plane partitions in $B(r,r,2t)$ is

The number of symmetric self-complementary plane partitions in $B(2r,2r,2t)$ is

in $B(2r+1,2r+1,2t)$ it is

The number of cyclically symmetric transpose complement plane partitions in $B(2r,2r,2r)$ is

The number of cyclically symmetric self-complementary plane partitions in $B(2r,2r,2r)$ is

The number of totally symmetric self-complementary plane partitions in $B(2r,2r,2r)$ is

A strict shifted plane partition is an arrangement of the parts in a partition so that each row is indented one space from the previous row and there is weak decrease across rows and strict decrease down columns. An example is given by:

A descending plane partition is a strict shifted plane partition in which the number of parts in each row is strictly less than the largest part in that row and is greater than or equal to the largest part in the next row. The example of a strict shifted plane partition also satisfies the conditions of a descending plane partition. The number of descending plane partitions in $B(r,r,r)$ is

The notation ${\sum }_{\pi \subseteq B(r,s,t)}$ denotes the sum over all plane partitions contained in $B(r,s,t)$, and $|\pi |$ denotes the number of elements in $\pi$.

where ${\sigma }_2(j)$ is the sum of the squares of the divisors of $j$.

As $n\to \mathrm{\infty }$

where $\zeta$ is the Riemann $\zeta$-function (§25.2(i)).

Addendum: To ten decimal places,