Inmathematics and especially incombinatorics, aplane partition is a two-dimensional array of nonnegative integers${\displaystyle {\pi }_{i,j}}$ (withpositiveinteger indicesi andj) that is nonincreasing in both indices. This means that

${\displaystyle {\pi }_{i,j}\ge {\pi }_{i,j+1}}$ and${\displaystyle {\pi }_{i,j}\ge {\pi }_{i+1,j}}$ for alli andj.

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

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 positiveinteger 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${\displaystyle 1\le i\le r}$,${\displaystyle 1\le j\le s}$, and${\displaystyle 1\le k\le t}$, then (i,j,k) also lies in${\displaystyle \mathcal{P}}$.

Thesum 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 theenumeration of plane partitions in various classes. The number of plane partitions with sumn is denoted by PL(n). For example, there are six plane partitions with sum 3

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.

Thegenerating function for PL(n) is^[1]

${\displaystyle \sum _{n=0}^{\mathrm{\infty }}\mathrm{PL}(n)x^n=\prod _{k=1}^{\mathrm{\infty }}\frac{1}{(1-x^k{)}^k}=1+x+3x^2+6x^3+13x^4+24x^5+\cdots }$(sequenceA000219 in theOEIS).

It is sometimes referred to as theMacMahon function, as it was discovered byPercy A. MacMahon.

This formula may be viewed as the 2-dimensional analogue ofEuler'sproduct formula for the number ofinteger partitions ofn. There is no analogous formula known for partitions in higher dimensions (i.e., forsolid partitions).^[2] The asymptotics for plane partitions were first calculated byE. M. Wright.^[3] One obtains, for large${\displaystyle n}$, that^[a]

${\displaystyle \mathrm{PL}(n)\sim \frac{\zeta (3{)}^{7/36}}{\sqrt{12\pi }}{\left(\frac{n}{2}\right)}^{-25/36}\exp \left(3\zeta (3{)}^{1/3}{\left(\frac{n}{2}\right)}^{2/3}+{\zeta }^{\prime }(-1)\right).}$

Evaluating numerically yields

${\displaystyle \ln \mathrm{PL}(n)\sim 2.00945n^{2/3}-0.69444\ln n-\mathrm{1.4631.}}$

Around 1896, MacMahon set up the generating function of plane partitions that are subsets of the${\displaystyle r\times s\times t}$ box${\displaystyle \mathcal{B}(r,s,t)=\{(i,j,k)|1\le i\le r,1\le j\le s,1\le k\le t\}}$ in his first paper on plane partitions.^[5] The formula is given by$${\displaystyle \sum _{\pi \in \mathcal{B}(r,s,t)}{q}^{|\pi |}=\prod _{i=1}^r\prod _{j=1}^s\frac{1-q^{i+j+t-1}}{1-q^{i+j-1}}}$$

A proof of this formula can be found in the bookCombinatory Analysis written by MacMahon.^[6] MacMahon also mentions the generating functions of plane partitions.^[7] The formula for the generating function can be written in an alternative way, which is given by$${\displaystyle \sum _{\pi \in \mathcal{B}(r,s,t)}{q}^{|\pi |}=\prod _{i=1}^r\prod _{j=1}^s\prod _{k=1}^t\frac{1-q^{i+j+k-1}}{1-q^{i+j+k-2}}}$$

Multiplying each component by${\displaystyle \frac{1-q}{1-q}}$, and settingq = 1 in the formulas above yields that the total number${\displaystyle N_1(r,s,t)}$ of plane partitions that fit in the${\displaystyle r\times s\times t}$ box${\displaystyle \mathcal{B}(r,s,t)}$ is equal to the following product formula:^[8]$${\displaystyle N_1(r,s,t)=\prod _{(i,j,k)\in \mathcal{B}(r,s,t)}\frac{i+j+k-1}{i+j+k-2}=\prod _{i=1}^r\prod _{j=1}^s\frac{i+j+t-1}{i+j-1}.}$$The planar case (whent = 1) yields thebinomial coefficients:

${\displaystyle \mathcal{B}(r,s,1)=(\genfrac{}{}{0ex}{}{r+s}{r}).}$

The general solution is

Theisometric projection of the unit cubes representing a plane partition in a box gives abijection between these plane partitions and rhombus tilings of a hexagon with the same edge lengths as the box.^[9]

Special plane partitions include symmetric, cyclic and self-complementary plane partitions, and combinations of these properties.

In the subsequent sections, the enumeration of special sub-classes of plane partitions inside a box are considered.These articles use the notation${\displaystyle N_i(r,s,t)}$ for the number of such plane partitions, wherer,s, andt are the dimensions of the box under consideration, andi is the index for the case being considered.

${\displaystyle {\mathcal{S}}_2}$ is the group ofpermutations acting on the first two coordinates of a point. This group contains the identity, which sends (i,j,k) to itself, and the transposition (i,j,k) → (j,i,k). The number of elements in an orbit${\displaystyle \eta }$ is denoted by${\displaystyle |\eta |}$.${\displaystyle \mathcal{B}/{\mathcal{S}}_2}$ denotes the set of orbits of elements of${\displaystyle \mathcal{B}}$ under the action of${\displaystyle {\mathcal{S}}_2}$. The height of an element (i,j,k) is defined by$${\displaystyle ht(i,j,k)=i+j+k-2.}$$The height increases by one for each step away from the back right corner. For example, the corner position (1, 1, 1) has height 1 andht(2, 1, 1) = 2. The height of an orbit is defined to be the height of any element in the orbit. This notation of the height differs from the notation ofIan G. Macdonald.^[10]

There is a natural action of the permutation group${\displaystyle {\mathcal{S}}_3}$ on a Ferrers diagram of a plane partition—this corresponds to simultaneously permuting the three coordinates of all nodes. This generalizes the conjugation operation forinteger partitions. The action of${\displaystyle {\mathcal{S}}_3}$ can generate new plane partitions starting from a given plane partition. Below there are shown six plane partitions of 4 that are generated by the${\displaystyle {\mathcal{S}}_3}$ action. Only the exchange of the first two coordinates is manifest in the representation given below.

${\displaystyle {\mathcal{C}}_3}$ is called the group of cyclic permutations and consists of

${\displaystyle (i,j,k)\to (i,j,k),(i,j,k)\to (j,k,i),\text{and }(i,j,k)\to (k,i,j).}$

A plane partition${\displaystyle \pi }$ is called symmetric ifπ_i,j =π_j,i for alli,j. In other words, a plane partition is symmetric if${\displaystyle (i,j,k)\in \mathcal{B}(r,s,t)}$ if and only if${\displaystyle (j,i,k)\in \mathcal{B}(r,s,t)}$. Plane partitions of this type are symmetric with respect to the planex =y. Below is an example of a symmetric plane partition and its visualisation.

In 1898, MacMahon formulated his conjecture about the generating function for symmetric plane partitions which are subsets of${\displaystyle \mathcal{B}(r,r,t)}$.^[11] This conjecture is called theMacMahon conjecture. The generating function is given by$${\displaystyle \sum _{\pi \in \mathcal{B}(r,r,t)/{\mathcal{S}}_2}{q}^{|\pi |}=\prod _{i=1}^r\left[\frac{1-q^{t+2i-1}}{1-q^{2i-1}}\prod _{j=i+1}^r\frac{1-q^{2(i+j+t-1)}}{1-q^{2(i+j-1)}}\right]}$$

Macdonald^[10] pointed out that Percy A. MacMahon's conjecture reduces to

${\displaystyle \sum _{\pi \in \mathcal{B}(r,r,t)/{\mathcal{S}}_2}{q}^{|\pi |}=\prod _{\eta \in \mathcal{B}(r,r,t)/{\mathcal{S}}_2}\frac{1-q^{|\eta |(1+ht(\eta ))}}{1-q^{|\eta |ht(\eta )}}}$

In 1972 Edward A. Bender andDonald E. Knuth conjectured^[12] a simple closed form for the generating function for plane partition which have at mostr rows and strict decrease along the rows.George Andrews showed^[13] that the conjecture of Bender and Knuth and the MacMahon conjecture are equivalent. MacMahon's conjecture was proven almost simultaneously by George Andrews in 1977^[14] and later Ian G. Macdonald presented an alternative proof.^[15] When settingq = 1 yields the counting function${\displaystyle N_2(r,r,t)}$ which is given by

For a proof of the caseq = 1 please refer to George Andrews' paperMacMahon's conjecture on symmetric plane partitions.^[16]

π is called cyclically symmetric, if thei-th row of${\displaystyle \pi }$ is conjugate to thei-th column for alli. Thei-th row is regarded as an ordinary partition. The conjugate of a partition${\displaystyle \pi }$ is the partition whose diagram is the transpose of partition${\displaystyle \pi }$.^[10] In other words, the plane partition is cyclically symmetric if whenever${\displaystyle (i,j,k)\in \mathcal{B}(r,s,t)}$ then (k,i,j) and (j,k,i) also belong to${\displaystyle \mathcal{B}(r,s,t)}$. Below an example of a cyclically symmetric plane partition and its visualization is given.

Macdonald's conjecture provides a formula for calculating the number of cyclically symmetric plane partitions for a given integerr. This conjecture is called theMacdonald conjecture. The generating function for cyclically symmetric plane partitions which are subsets of${\displaystyle \mathcal{B}(r,r,r)}$ is given by

${\displaystyle \sum _{\pi \in \mathcal{B}(r,r,r)/{\mathcal{C}}_3}{q}^{|\pi |}=\prod _{\eta \in \mathcal{B}(r,r,r)/{\mathcal{C}}_3}\frac{1-q^{|\eta |(1+ht(\eta ))}}{1-q^{|\eta |ht(\eta )}}}$

This equation can also be written in another way

${\displaystyle \prod _{\eta \in \mathcal{B}(r,r,r)/{\mathcal{C}}_3}\frac{1-q^{|\eta |(1+ht(\eta ))}}{1-q^{|\eta |ht(\eta )}}=\prod _{i=1}^r\left[\frac{1-q^{3i-1}}{1-q^{3i-2}}\prod _{j=i}^r\frac{1-q^{3(r+i+j-1)}}{1-q^{3(2i+j-1)}}\right]}$

In 1979, Andrews proved Macdonald's conjecture for the caseq = 1 as the"weak" Macdonald conjecture.^[17] Three years later William H. Mills,David Robbins and Howard Rumsey proved the general case of Macdonald's conjecture in their paperProof of the Macdonald conjecture.^[18] The formula for${\displaystyle N_3(r,r,r)}$ is given by the"weak" Macdonald conjecture

${\displaystyle N_3(r,r,r)=\prod _{i=1}^r\left[\frac{3i-1}{3i-2}\prod _{j=i}^r\frac{i+j+r-1}{2i+j-1}\right]}$

A totally symmetric plane partition${\displaystyle \pi }$ is a plane partition which is symmetric and cyclically symmetric. This means that the diagram is symmetric at all three diagonal planes, or in other words that if${\displaystyle (i,j,k)\in \mathcal{B}(r,s,t)}$ then all six permutations of (i,j,k) are also in${\displaystyle \mathcal{B}(r,s,t)}$. Below an example of a matrix for a totally symmetric plane partition is given. The picture shows the visualisation of the matrix.

Macdonald found the total number of totally symmetric plane partitions that are subsets of${\displaystyle \mathcal{B}(r,r,r)}$. The formula is given by

${\displaystyle N_4(r,r,r)=\prod _{\eta \in \mathcal{B}(r,r,r)/{\mathcal{S}}_3}\frac{1+ht(\eta )}{ht(\eta )}}$

In 1995John R. Stembridge first proved the formula for${\displaystyle N_4(r,r,r)}$^[19] and later in 2005 it was proven by George Andrews,Peter Paule, and Carsten Schneider.^[20] Around 1983 Andrews and Robbins independently stated an explicit product formula for the orbit-counting generating function for totally symmetric plane partitions.^[21][22] This formula already alluded to in George E. Andrews' paperTotally symmetric plane partitions which was published 1980.^[23] The conjecture is calledTheq-TSPPconjecture and it is given by:

Let${\displaystyle {\mathcal{S}}_3}$ be the symmetric group. The orbit counting function for totally symmetric plane partitions that fit inside${\displaystyle \mathcal{B}(r,r,r)}$ is given by the formula

${\displaystyle \sum _{\pi \in \mathcal{B}(r,r,r)/{\mathcal{S}}_3}{q}^{|\pi |}=\prod _{\eta \in \mathcal{B}(r,r,r)/{\mathcal{S}}_3}\frac{1-q^{1+ht(\eta )}}{1-q^{ht(\eta )}}=\prod _{1\le i\le j\le k\le r}\frac{1-q^{i+j+k-1}}{1-q^{i+j+k-2}}.}$

This conjecture was proved in 2011 byChristoph Koutschan,Manuel Kauers andDoron Zeilberger.^[24]

If${\displaystyle {\pi }_{i,j}+{\pi }_{r-i+1,s-j+1}=t}$ for all${\displaystyle 1\le i\le r}$,${\displaystyle 1\le j\le s}$, then the plane partition is called self-complementary. It is necessary that the product${\displaystyle r\cdot s\cdot t}$ is even. Below an example of a self-complementary symmetric plane partition and its visualisation is given.

Richard P. Stanley^[25] conjectured formulas for the total number of self-complementary plane partitions${\displaystyle N_5(r,s,t)}$. According to Stanley, Robbins also formulated formulas for the total number of self-complementary plane partitions in a different but equivalent form. The total number of self-complementary plane partitions that are subsets of${\displaystyle \mathcal{B}(r,s,t)}$ is given by

${\displaystyle N_5(2r,2s,2t)=N_1(r,s,t{)}^2}$

${\displaystyle N_5(2r+1,2s,2t)=N_1(r,s,t)N_1(r+1,s,t)}$

${\displaystyle N_5(2r+1,2s+1,2t)=N_1(r+1,s,t)N_1(r,s+1,t)}$

It is necessary that the product ofr,s andt is even. A proof can be found in the paperSymmetries of Plane Partitions which was written by Stanley.^[26][25] The proof works with Schur functions${\displaystyle s_{s^r}(x)}$. Stanley's proof of the ordinary enumeration of self-complementary plane partitions yields theq-analogue by substituting${\displaystyle x_i=q^i}$ for${\displaystyle i=1,\dots ,n}$.^[27] This is a special case of Stanley's hook-content formula.^[28] The generating function for self-complementary plane partitions is given by

${\displaystyle s_{{\gamma }^{\alpha }}(q,q^2,\dots ,q^n)=q^{\gamma \alpha (\alpha +1)/2}\prod _{i=1}^{\alpha }\prod _{j=0}^{\gamma -1}\frac{1-q^{i+n-\alpha +j}}{1-q^{i+j}}}$

Substituting this formula in

${\displaystyle s_{s^r}(x_1,x_2,\dots ,x_{t+r}{)}^2\text{ for }\mathcal{B}(2r,2s,2t)}$

${\displaystyle s_{s^r}(x_1,x_2,\dots ,x_{t+r})s_{(s+1{)}^r}(x_1,x_2,\dots ,x_{t+r})\text{ for }\mathcal{B}(2r,2s+1,2t)}$

${\displaystyle s_{s^{r+1}}(x_1,x_2,\dots ,x_{t+r+1})s_{s^r}(x_1,x_2,\dots ,x_{t+r+1})\text{ for }\mathcal{B}(2r+1,2s,2t+1)}$

supplies the desiredq-analogue case.

A plane partition${\displaystyle \pi }$ is called cyclically symmetric self-complementary if it iscyclically symmetric andself-complementary. The figure presents a cyclically symmetric self-complementary plane partition and the according matrix is below.

In a private communication with Stanley, Robbins conjectured that the total number of cyclically symmetric self-complementary plane partitions is given by${\displaystyle N_6(2r,2r,2r)}$.^[22][25] The total number of cyclically symmetric self-complementary plane partitions is given by

${\displaystyle N_6(2r,2r,2r)=D_r^2}$

${\displaystyle D_r}$ is the number of${\displaystyle r\times r}$alternating sign matrices. A formula for${\displaystyle D_r}$ is given by

${\displaystyle D_r=\prod _{j=0}^{r-1}\frac{(3j+1)!}{(r+j)!}}$

Greg Kuperberg proved the formula for${\displaystyle N_6(r,r,r)}$ in 1994.^[9]

A totally symmetric self-complementary plane partition is a plane partition that is bothtotally symmetric andself-complementary. For instance, the matrix below is such a plane partition; it is visualised in the accompanying picture.

The formula${\displaystyle N_7(r,r,r)}$ was conjectured by William H. Mills, Robbins and Howard Rumsey in their workSelf-Complementary Totally Symmetric Plane Partitions.^[29] The total number of totally symmetric self-complementary plane partitions is given by

${\displaystyle N_7(2r,2r,2r)=D_r}$

Andrews proves this formula in 1994 in his paperPlane Partitions V: The TSSCPP Conjecture.^[30]

• Gaussian binomial coefficients
• Voxel

• G. Andrews,The Theory of Partitions, Cambridge University Press, Cambridge, 1998,ISBN 0-521-63766-X
• Bender, Edward A.;Knuth, Donald E. (1972), "Enumeration of plane partitions",Journal of Combinatorial Theory, Series A,13:40–54,doi:10.1016/0097-3165(72)90007-6,ISSN 1096-0899,MR 0299574
• I.G. Macdonald,Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, 1999,ISBN 0-19-850450-0
• P.A. MacMahon,Combinatory analysis, 2 vols, Cambridge University Press, 1915–16.

• Weisstein, Eric W."Plane partition".MathWorld.
• The DLMF page on Plane Partitions

• Enumerative combinatorics
• Integer partitions

• Articles with short description
• Short description is different from Wikidata