26 Combinatorial Analysis Properties 26.11 Integer Partitions: Compositions 26.13 Permutations: Cycle Notation

§26.12 Plane Partitions

Permalink:: http://dlmf.nist.gov/26.12

§26.12(i) Definitions
§26.12(ii) Generating Functions
§26.12(iii) Recurrence Relation
§26.12(iv) Limiting Form

§26.12(i) Definitions

Notes:: See Bressoud (1999, pp. 197–199) (as corrected here). Table 26.12.1 was computed by the author. For (26.12.7) see Andrews (1979, p. 195).
Defines:: $\mathop{\mathit{pp}\/}\nolimits\!\left(n\right)$ : number of plane partitions of
Keywords:: box, plane partitions
Permalink:: http://dlmf.nist.gov/26.12.i

A plane partition, $\pi$ , of a positive integer , is a partition of 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

$\begin{array}[]{cc}2&1\end{array}$ ,

$\begin{array}[]{c}2\\ 1\end{array}$ ,

$\begin{array}[]{ccc}1&1&1\end{array}$ ,

$\begin{array}[]{cc}1&1\\ 1\end{array}$ ,

$\begin{array}[]{c}1\\ 1\\ 1\end{array}$ .

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

An equivalent definition is that a plane partition is a finite subset of $\NatNumber\times\NatNumber\times\NatNumber$ with the property that if $(r,s,t)\in\pi$ and $(1,1,1)\leq(h,j,k)\leq(r,s,t)$ , then 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).

Figure 26.12.1: A plane partition of 75.

Referenced by:: §26.12(i), §26.12(i)
Permalink:: http://dlmf.nist.gov/26.12.F1
Encodings:: pdf, png

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

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

Table 26.12.1: Plane partitions.

	$\mathop{\mathit{pp}\/}\nolimits\!\left(n\right)$		$\mathop{\mathit{pp}\/}\nolimits\!\left(n\right)$		$\mathop{\mathit{pp}\/}\nolimits\!\left(n\right)$
0	1	17	18334	34	281 75955
1	1	18	29601	35	416 91046
2	3	19	47330	36	614 84961
3	6	20	75278	37	903 79784
4	13	21	1 18794	38	1324 41995
5	24	22	1 86475	39	1934 87501
6	48	23	2 90783	40	2818 46923
7	86	24	4 51194	41	4093 83981
8	160	25	6 96033	42	5930 01267
9	282	26	10 68745	43	8566 67495
10	500	27	16 32658	44	12343 63833
11	859	28	24 83234	45	17740 79109
12	1479	29	37 59612	46	25435 35902
13	2485	30	56 68963	47	36379 93036
14	4167	31	85 12309	48	51913 04973
15	6879	32	127 33429	49	73910 26522
16	11297	33	189 74973	50	1 04996 40707

Symbols:: $\mathop{\mathit{pp}\/}\nolimits\!\left(n\right)$ : number of plane partitions of and : nonnegative integer
Keywords:: partitions, plane partitions
Referenced by:: §26.12(i), §26.12(i)
Permalink:: http://dlmf.nist.gov/26.12.T1

We define the $r\times s\times t$ box as

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

Symbols:: : positive integer, : positive integer, : positive integer, : positive integer, : positive integer, : positive integer and : Box
Permalink:: http://dlmf.nist.gov/26.12.E3
Encodings:: TeX, pMML, png

Then the number of plane partitions in is

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

Symbols:: $\in$ : element of, : positive integer, : positive integer, : positive integer, : positive integer, : positive integer, : positive integer and : Box
Permalink:: http://dlmf.nist.gov/26.12.E4
Encodings:: TeX, pMML, png

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 is

26.12.5 $\prod_{{h=1}}^{r}\frac{2h+t-1}{2h-1}\prod_{{1\leq h<j\leq r}}\frac{h+j+t-1}{h+% j-1}.$

Symbols:: : positive integer, : positive integer, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/26.12.E5
Encodings:: TeX, pMML, png

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 is

26.12.6 $\prod_{{h=1}}^{r}\frac{3h-1}{3h-2}\prod_{{1\leq h<j\leq r}}\frac{h+2j-1}{h+j-1},$

Symbols:: : positive integer, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/26.12.E6
Encodings:: TeX, pMML, png

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:: : positive integer, : positive integer and : positive integer
Referenced by:: §26.12(i)
Permalink:: http://dlmf.nist.gov/26.12.E7
Encodings:: TeX, pMML, png

A plane partition is totally symmetric if it is both symmetric and cyclically symmetric. The number of totally symmetric plane partitions in is

26.12.8 $\prod_{{1\leq h\leq j\leq r}}\frac{h+j+r-1}{h+2j-2}.$

Symbols:: : positive integer, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/26.12.E8
Encodings:: TeX, pMML, png

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 is

26.12.9 $\left(\prod_{{h=1}}^{r}\prod_{{j=1}}^{s}\frac{h+j+t-1}{h+j-1}\right)^{2};$

Symbols:: : positive integer, : positive integer, : positive integer, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/26.12.E9
Encodings:: TeX, pMML, png

in it is

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

Symbols:: : positive integer, : positive integer, : positive integer, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/26.12.E10
Encodings:: TeX, pMML, png

in it is

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

Symbols:: : positive integer, : positive integer, : positive integer, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/26.12.E11
Encodings:: TeX, pMML, png

A plane partition is transpose complement if it is equal to the reflection through the -plane of its complement. The number of transpose complement plane partitions in is

26.12.12 $\binom{t+r-1}{r-1}\prod_{{1\leq h\leq j\leq r-2}}\frac{h+j+2t+1}{h+j+1}.$

Symbols:: $\binom{m}{n}$ : binomial coefficient, : positive integer, : positive integer, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/26.12.E12
Encodings:: TeX, pMML, png

The number of symmetric self-complementary plane partitions in is

26.12.13 $\prod_{{h=1}}^{r}\prod_{{j=1}}^{r}\frac{h+j+t-1}{h+j-1};$

Symbols:: : positive integer, : positive integer, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/26.12.E13
Encodings:: TeX, pMML, png

in it is

26.12.14 $\prod_{{h=1}}^{{r}}\prod_{{j=1}}^{{r+1}}\frac{h+j+t-1}{h+j-1}.$

Symbols:: : positive integer, : positive integer, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/26.12.E14
Encodings:: TeX, pMML, png

The number of cyclically symmetric transpose complement plane partitions in is

26.12.15 $\prod_{{h=0}}^{{r-1}}\frac{(3h+1)\,(6h)!\,(2h)!}{(4h+1)!\,(4h)!}.$

Symbols:: : : factorial, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/26.12.E15
Encodings:: TeX, pMML, png

The number of cyclically symmetric self-complementary plane partitions in is

26.12.16 $\left(\prod_{{h=0}}^{{r-1}}\frac{(3h+1)!}{(r+h)!}\right)^{2}.$

Symbols:: : : factorial, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/26.12.E16
Encodings:: TeX, pMML, png

The number of totally symmetric self-complementary plane partitions in is

26.12.17 $\prod_{{h=0}}^{{r-1}}\frac{(3h+1)!}{(r+h)!}.$

Symbols:: : : factorial, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/26.12.E17
Encodings:: TeX, pMML, png

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

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 is

26.12.19 $\prod_{{h=0}}^{{r-1}}\frac{(3h+1)!}{(r+h)!}.$

Symbols:: : : factorial, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/26.12.E19
Encodings:: TeX, pMML, png

§26.12(ii) Generating Functions

Notes:: See Bressoud (1999, pp. 11, 13–18, 22). For (26.12.22) see also Andrews (1979, p. 195).
Keywords:: plane partitions
Permalink:: http://dlmf.nist.gov/26.12.ii

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

26.12.20 $\sum_{{\pi\subseteq\NatNumber\times\NatNumber\times\NatNumber}}q^{{|\pi|}}=% \prod_{{k=1}}^{{\infty}}\frac{1}{(1-q^{k})^{k}},$

Symbols:: $\NatNumber$ : set of all positive integers, $\subseteq$ : is, or is contained in, : nonnegative integer, $\pi$ : plane partition and : parameter
Permalink:: http://dlmf.nist.gov/26.12.E20
Encodings:: TeX, pMML, png

26.12.21 $\sum_{{\pi\subseteq B(r,s,t)}}q^{{|\pi|}}=\prod_{{(h,j,k)\in B(r,s,t)}}\frac{1% -q^{{h+j+k-1}}}{1-q^{{h+j+k-2}}}=\prod_{{h=1}}^{r}\prod_{{j=1}}^{s}\frac{1-q^{% {h+j+t-1}}}{1-q^{{h+j-1}}},$

Symbols:: $\in$ : element of, $\subseteq$ : is, or is contained in, : nonnegative integer, : nonnegative integer, : nonnegative integer, $\pi$ : plane partition, : positive integer, : positive integer, : positive integer, : Box and : parameter
Permalink:: http://dlmf.nist.gov/26.12.E21
Encodings:: TeX, pMML, png

26.12.22 $\sum_{{\substack{\pi\subseteq B(r,r,t)\\ \pi\mbox{\scriptsize\ symmetric}}}}q^{{|\pi|}}=\prod_{{h=1}}^{r}\frac{1-q^{{2h% +t-1}}}{1-q^{{2h-1}}}\prod_{{1\leq h<j\leq r}}\frac{1-q^{{2(h+j+t-1)}}}{1-q^{{% 2(h+j-1)}}}.$

Symbols:: $\subseteq$ : is, or is contained in, : nonnegative integer, : nonnegative integer, $\pi$ : plane partition, : positive integer, : positive integer, : Box and : parameter
Referenced by:: §26.12(ii)
Permalink:: http://dlmf.nist.gov/26.12.E22
Encodings:: TeX, pMML, png

26.12.23 $\sum_{{\substack{\pi\subseteq B(r,r,r)\\ \pi\mbox{\scriptsize\ cyclically symmetric}}}}q^{{\left|\pi\right|}}=\prod_{{h% =1}}^{r}\frac{1-q^{{3h-1}}}{1-q^{{3h-2}}}\prod_{{1\leq h<j\leq r}}\frac{1-q^{{% 3(h+2j-1)}}}{1-q^{{3(h+j-1)}}}=\prod_{{h=1}}^{r}\left(\frac{1-q^{{3h-1}}}{1-q^% {{3h-2}}}\prod_{{j=h}}^{r}\frac{1-q^{{3(r+h+j-1)}}}{1-q^{{3(2h+j-1)}}}\right).$

Symbols:: $\subseteq$ : is, or is contained in, : nonnegative integer, : nonnegative integer, $\pi$ : plane partition, : positive integer, : Box and : parameter
Permalink:: http://dlmf.nist.gov/26.12.E23
Encodings:: TeX, pMML, png

26.12.24 $\sum_{{\substack{\pi\subseteq B(r,r,r)\\ \pi\mbox{\scriptsize\ descending plane partition}}}}q^{{|\pi|}}=\prod_{{1\leq h% <j\leq r}}\frac{1-q^{{r+h+j-1}}}{1-q^{{2h+j-1}}}.$

Symbols:: $\subseteq$ : is, or is contained in, : nonnegative integer, : nonnegative integer, $\pi$ : plane partition, : positive integer, : Box and : parameter
Permalink:: http://dlmf.nist.gov/26.12.E24
Encodings:: TeX, pMML, png

§26.12(iii) Recurrence Relation

Notes:: See Bressoud (1999, p. 57).
Keywords:: plane partitions
Permalink:: http://dlmf.nist.gov/26.12.iii

26.12.25 $\mathop{\mathit{pp}\/}\nolimits\!\left(n\right)=\frac{1}{n}\sum_{{j=1}}^{n}% \mathop{\mathit{pp}\/}\nolimits\!\left(n-j\right)\sigma_{2}(j),$

Symbols:: $\mathop{\mathit{pp}\/}\nolimits\!\left(n\right)$ : number of plane partitions of , : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.12.E25
Encodings:: TeX, pMML, png

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

§26.12(iv) Limiting Form

Notes:: See Mutafchiev and Kamenov (2006). (A former incorrect version of (26.12.26) was from Andrews (1976, p. 199)).
Keywords:: partitions, plane partitions
Permalink:: http://dlmf.nist.gov/26.12.iv
Addition (effective with 1.0.5):: Mutafchiev and Kamenov (2006) was added as the reference for this subsection, replacing Andrews (1976).
Reported 2011-09-05

As $n\to\infty$

26.12.26 $\mathop{\mathit{pp}\/}\nolimits\!\left(n\right)\sim\frac{\left(\mathop{\zeta\/% }\nolimits\!\left(3\right)\right)^{{7/36}}}{2^{{11/36}}(3\pi)^{{1/2}}n^{{25/36% }}}\mathop{\exp\/}\nolimits\left(3\left(\mathop{\zeta\/}\nolimits\!\left(3% \right)\right)^{{1/3}}\left(\tfrac{1}{2}n\right)^{{2/3}}+{\mathop{\zeta\/}% \nolimits^{{\prime}}}\!\left(-1\right)\right),$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathop{\mathit{pp}\/}\nolimits\!\left(n\right)$ : number of plane partitions of , $\sim$ : asymptotic equality, : nonnegative integer and $\pi$ : plane partition
Referenced by:: §26.12(iv), 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, $\mathop{\mathit{pp}\/}\nolimits\!\left(n\right)\sim\left(\frac{\mathop{\zeta\/% }\nolimits\!\left(3\right)}{2^{{11}}n^{{25}}}\right)^{{1/36}}\*\mathop{\exp\/}% \nolimits\left(3\left(\frac{\mathop{\zeta\/}\nolimits\!\left(3\right)n^{2}}{4}% \right)^{{1/3}}+{\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(-1\right)\right),$ has been corrected.
Reported 2011-09-05 by Suresh Govindarajan

where $\mathop{\zeta\/}\nolimits$ is the Riemann $\mathop{\zeta\/}\nolimits$ -function (§25.2(i)).

Addendum: To ten decimal places,

26.12.27

$\mathop{\zeta\/}\nolimits\!\left(3\right)=1.20205\;69032,$

${\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(-1\right)=-0.16542\;11437.$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function
Referenced by:: Other Changes
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 $\mathop{\zeta\/}\nolimits\!\left(3\right)$ and ${\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(-1\right)$ . It will be included in the next print edition.
Reported 2012-04-12

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 26.11 Integer Partitions: Compositions 26.13 Permutations: Cycle Notation

§26.12 Plane Partitions

Contents

§26.12(i) Definitions

§26.12(ii) Generating Functions

§26.12(iii) Recurrence Relation

§26.12(iv) Limiting Form