# §26.12 Plane Partitions

## §26.12(i) Definitions

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:

 26.12.1 $3$, $\begin{array}[]{cc}2&1\end{array}$, $\begin{array}[]{c}2\\ 1\end{array}$, $\begin{array}[]{ccc}1&1&1\end{array}$, $\begin{array}[]{cc}1&1\\ 1\end{array}$, $\begin{array}[]{c}1\\ 1\\ 1\end{array}$. ⓘ Permalink: http://dlmf.nist.gov/26.12.E1 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png See also: Annotations for §26.12(i), §26.12 and Ch.26

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 $(1,1,1)\leq(h,j,k)\leq(r,s,t)$, then $(h,j,k)$ must be an element of $\pi$. Here $(h,j,k)\leq(r,s,t)$ means $h\leq r$, $j\leq s$, and $k\leq t$. 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).

 26.12.2 $\begin{array}[]{cccccc}6&5&5&4&3&3\\ 6&4&3&3&1\\ 6&4&3&1&1\\ 4&2&2&1\\ 3&1&1\\ 1&1&1\end{array}$ ⓘ Referenced by: §26.12(i) Permalink: http://dlmf.nist.gov/26.12.E2 Encodings: TeX, pMML, png See also: Annotations for §26.12(i), §26.12 and Ch.26

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

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

 26.12.3 $B(r,s,t)=\{(h,j,k)\>|\>1\leq h\leq r,1\leq j\leq s,1\leq k\leq t\}.$

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

 26.12.4 $\prod_{(h,j,k)\in B(r,s,t)}\frac{h+j+k-1}{h+j+k-2}=\prod_{h=1}^{r}\prod_{j=1}^% {s}\frac{h+j+t-1}{h+j-1}.$

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

 26.12.5 $\prod_{h=1}^{r}\frac{2h+t-1}{2h-1}\prod_{1\leq h ⓘ Symbols: $r$: positive integer, $t$: positive integer, $h$: positive integer and $j$: positive integer Permalink: http://dlmf.nist.gov/26.12.E5 Encodings: TeX, pMML, png See also: Annotations for §26.12(i), §26.12 and Ch.26

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

 26.12.6 $\prod_{h=1}^{r}\frac{3h-1}{3h-2}\prod_{1\leq h ⓘ Symbols: $r$: positive integer, $h$: positive integer and $j$: positive integer Permalink: http://dlmf.nist.gov/26.12.E6 Encodings: TeX, pMML, png See also: Annotations for §26.12(i), §26.12 and Ch.26

or equivalently,

 26.12.7 $\prod_{h=1}^{r}\left(\frac{3h-1}{3h-2}\prod_{j=h}^{r}\frac{r+h+j-1}{2h+j-1}% \right).$ ⓘ Symbols: $r$: positive integer, $h$: positive integer and $j$: positive integer Referenced by: §26.12(i) Permalink: http://dlmf.nist.gov/26.12.E7 Encodings: TeX, pMML, png See also: Annotations for §26.12(i), §26.12 and Ch.26

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

 26.12.8 $\prod_{1\leq h\leq j\leq r}\frac{h+j+r-1}{h+2j-2}.$ ⓘ Symbols: $r$: positive integer, $h$: positive integer and $j$: positive integer Permalink: http://dlmf.nist.gov/26.12.E8 Encodings: TeX, pMML, png See also: Annotations for §26.12(i), §26.12 and Ch.26

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

 26.12.9 $\left(\prod_{h=1}^{r}\prod_{j=1}^{s}\frac{h+j+t-1}{h+j-1}\right)^{2};$ ⓘ Symbols: $r$: positive integer, $s$: positive integer, $t$: positive integer, $h$: positive integer and $j$: positive integer Permalink: http://dlmf.nist.gov/26.12.E9 Encodings: TeX, pMML, png See also: Annotations for §26.12(i), §26.12 and Ch.26

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

 26.12.10 $\left(\prod_{h=1}^{r}\prod_{j=1}^{s}\frac{h+j+t-1}{h+j-1}\right)\*\left(\prod_% {h=1}^{r+1}\prod_{j=1}^{s}\frac{h+j+t-1}{h+j-1}\right);$ ⓘ

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

 26.12.11 $\left(\prod_{h=1}^{r+1}\prod_{j=1}^{s}\frac{h+j+t-1}{h+j-1}\right)\*\left(% \prod_{h=1}^{r}\prod_{j=1}^{s+1}\frac{h+j+t-1}{h+j-1}\right).$ ⓘ

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

 26.12.12 $\genfrac{(}{)}{0.0pt}{}{t+r-1}{r-1}\prod_{1\leq h\leq j\leq r-2}\frac{h+j+2t+1% }{h+j+1}.$

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

 26.12.13 $\prod_{h=1}^{r}\prod_{j=1}^{r}\frac{h+j+t-1}{h+j-1};$ ⓘ Symbols: $r$: positive integer, $t$: positive integer, $h$: positive integer and $j$: positive integer Permalink: http://dlmf.nist.gov/26.12.E13 Encodings: TeX, pMML, png See also: Annotations for §26.12(i), §26.12 and Ch.26

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

 26.12.14 $\prod_{h=1}^{r}\prod_{j=1}^{r+1}\frac{h+j+t-1}{h+j-1}.$ ⓘ Symbols: $r$: positive integer, $t$: positive integer, $h$: positive integer and $j$: positive integer Permalink: http://dlmf.nist.gov/26.12.E14 Encodings: TeX, pMML, png See also: Annotations for §26.12(i), §26.12 and Ch.26

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

 26.12.15 $\prod_{h=0}^{r-1}\frac{(3h+1)\,(6h)!\,(2h)!}{(4h+1)!\,(4h)!}.$ ⓘ Symbols: $!$: factorial (as in $n!$), $r$: positive integer and $h$: positive integer Permalink: http://dlmf.nist.gov/26.12.E15 Encodings: TeX, pMML, png See also: Annotations for §26.12(i), §26.12 and Ch.26

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

 26.12.16 $\left(\prod_{h=0}^{r-1}\frac{(3h+1)!}{(r+h)!}\right)^{2}.$ ⓘ Symbols: $!$: factorial (as in $n!$), $r$: positive integer and $h$: positive integer Permalink: http://dlmf.nist.gov/26.12.E16 Encodings: TeX, pMML, png See also: Annotations for §26.12(i), §26.12 and Ch.26

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

 26.12.17 $\prod_{h=0}^{r-1}\frac{(3h+1)!}{(r+h)!}.$ ⓘ Symbols: $!$: factorial (as in $n!$), $r$: positive integer and $h$: positive integer Permalink: http://dlmf.nist.gov/26.12.E17 Encodings: TeX, pMML, png See also: Annotations for §26.12(i), §26.12 and Ch.26

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:

 26.12.18 $\begin{array}[]{ccccc}6&6&6&4&3\\ &3&3\\ &&2\end{array}$ ⓘ Permalink: http://dlmf.nist.gov/26.12.E18 Encodings: TeX, pMML, png See also: Annotations for §26.12(i), §26.12 and Ch.26

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

 26.12.19 $\prod_{h=0}^{r-1}\frac{(3h+1)!}{(r+h)!}.$ ⓘ Symbols: $!$: factorial (as in $n!$), $r$: positive integer and $h$: positive integer Permalink: http://dlmf.nist.gov/26.12.E19 Encodings: TeX, pMML, png See also: Annotations for §26.12(i), §26.12 and Ch.26

## §26.12(ii) Generating Functions

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 $\pi$.

 26.12.20 $\sum_{\pi\subseteq\mathbb{N}\times\mathbb{N}\times\mathbb{N}}q^{|\pi|}=\prod_{% k=1}^{\infty}\frac{1}{(1-q^{k})^{k}},$
 26.12.21 $\sum_{\pi\subseteq B(r,s,t)}q^{|\pi|}=\prod_{(h,j,k)\in B(r,s,t)}\frac{1-q^{h+% j+k-1}}{1-q^{h+j+k-2}}=\prod_{h=1}^{r}\prod_{j=1}^{s}\frac{1-q^{h+j+t-1}}{1-q^% {h+j-1}},$
 26.12.22 $\sum_{\begin{subarray}{c}\pi\subseteq B(r,r,t)\\ \pi\mbox{\scriptsize\ symmetric}\end{subarray}}q^{|\pi|}=\prod_{h=1}^{r}\frac{% 1-q^{2h+t-1}}{1-q^{2h-1}}\prod_{1\leq h
 26.12.23 $\sum_{\begin{subarray}{c}\pi\subseteq B(r,r,r)\\ \pi\mbox{\scriptsize\ cyclically symmetric}\end{subarray}}q^{\left|\pi\right|}% =\prod_{h=1}^{r}\frac{1-q^{3h-1}}{1-q^{3h-2}}\prod_{1\leq h
 26.12.24 $\sum_{\begin{subarray}{c}\pi\subseteq B(r,r,r)\\ \pi\mbox{\scriptsize\ descending plane partition}\end{subarray}}q^{|\pi|}=% \prod_{1\leq h

## §26.12(iii) Recurrence Relation

 26.12.25 $\operatorname{pp}\left(n\right)=\frac{1}{n}\sum_{j=1}^{n}\operatorname{pp}% \left(n-j\right)\sigma_{2}(j),$ ⓘ Symbols: $\operatorname{pp}\left(\NVar{n}\right)$: number of plane partitions of $n$, $j$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/26.12.E25 Encodings: TeX, pMML, png See also: Annotations for §26.12(iii), §26.12 and Ch.26

where $\sigma_{2}(j)$ is the sum of the squares of the divisors of $j$.

## §26.12(iv) Limiting Form

As $n\to\infty$

 26.12.26 $\operatorname{pp}\left(n\right)\sim\frac{\left(\zeta\left(3\right)\right)^{7/3% 6}}{2^{11/36}(3\pi)^{1/2}n^{25/36}}\exp\left(3\left(\zeta\left(3\right)\right)% ^{1/3}\left(\tfrac{1}{2}n\right)^{2/3}+\zeta'\left(-1\right)\right),$ ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\sim$: asymptotic equality, $\exp\NVar{z}$: exponential function, $\operatorname{pp}\left(\NVar{n}\right)$: number of plane partitions of $n$, $n$: nonnegative integer and $\pi$: plane partition Referenced by: §26.12(iv), Erratum (V1.0.5) for Equation (26.12.26) Permalink: http://dlmf.nist.gov/26.12.E26 Encodings: TeX, pMML, png Errata (effective with 1.0.5): The original statement of this equation, $\operatorname{pp}\left(n\right)\sim\left(\frac{\zeta\left(3\right)}{2^{11}n^{2% 5}}\right)^{1/36}\*\exp\left(3\left(\frac{\zeta\left(3\right)n^{2}}{4}\right)^% {1/3}+\zeta'\left(-1\right)\right)$, has been corrected. Reported 2011-09-05 by Suresh Govindarajan See also: Annotations for §26.12(iv), §26.12 and Ch.26

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

 26.12.27 $\displaystyle\zeta\left(3\right)$ $\displaystyle=1.20205\;69032,$ $\displaystyle\zeta'\left(-1\right)$ $\displaystyle=-0.16542\;11437.$ ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function Referenced by: Erratum (V1.0.5) for Chapters 8, 20, 36 Permalink: http://dlmf.nist.gov/26.12.E27 Encodings: TeX, TeX, pMML, pMML, png, png Addition (effective with 1.0.5): This equation has been added to give decimal values of $\zeta\left(3\right)$ and $\zeta'\left(-1\right)$. It will be included in the next print edition. See also: Annotations for §26.12(iv), §26.12 and Ch.26