# Plane partition

A plane partition represented as piles of unit cubes

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

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

Moreover only finitely many of the ${\displaystyle \pi _{i,j}}$ are nonzero. A plane partitions 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 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. The number of plane partitions with sum n is denoted PL(n).

For example, there are six plane partitions with sum 3:

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

so PL(3) = 6. (Here the plane partitions are drawn using matrix indexing for the coordinates and the entries equal to 0 are suppressed for readability.) Let ${\displaystyle N_{i}(r,s,t)}$ be the total number of plane partitions in which r is the number of rows that are unequal to zero, s is the number of columns that are non-zero, and t is the largest integer of the matrix. Plane partitions are often described by the positions of the unit cubes. Therefore a plane partition is 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 (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 ${\displaystyle {\mathcal {P}}}$.

${\displaystyle {\mathcal {B}}(r,s,t)=\{(i,j,k)|1\leq i\leq r,1\leq j\leq s,1\leq k\leq t\}}$

## Generating function of plane partitions

By a result of Percy A. MacMahon, the generating function for PL(n) is given by

${\displaystyle \sum _{n=0}^{\infty }{\mbox{PL}}(n)\,x^{n}=\prod _{k=1}^{\infty }{\frac {1}{(1-x^{k})^{k}}}=1+x+3x^{2}+6x^{3}+13x^{4}+24x^{5}+\cdots .}$[1]

This is sometimes referred to as the MacMahon function.

This formula may be viewed as the 2-dimensional analogue of Euler's product formula for the number of integer partitions of n. There is no analogous formula known for partitions in higher dimensions (i.e., for solid partitions).[2] The asymptotics of plane partitions was worked out by E. M. Wright.[3] One obtains, for large ${\displaystyle n}$:

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

Here the typographical error (in Wright's paper) has been corrected, pointed out by Mutafchiev and Kamenov.[4] Evaluating numerically yields

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

Around 1896 Percy A. MacMahon set up the generating function of plane partitions that are subsets of ${\displaystyle {\mathcal {B}}(r,s,t)}$ 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 book Combinatory Analysis written by Percy A. MacMahon.[6] Percy A. MacMahon also mentions in his book Combinatory Analysis the generating functions of plane partitions in article 429.[7] The formula for the generating function can be written in an alternative way, which is given by

${\displaystyle \sum _{\pi \in {\mathcal {B}}(r,s,t)}q^{|\pi |}=\prod _{i=1}^{r}\prod _{j=1}^{s}\prod _{k=1}^{t}{\frac {1-q^{i+j+k-1}}{1-q^{i+j+k-2}}}}$

Setting q=1 in the formulas above yields

${\displaystyle N_{1}(r,s,t)=\prod _{(i,j,k)\in {\mathcal {B}}(r,s,t)}{\frac {i+j+k-1}{i+j+k-2}}=\prod _{i=1}^{r}\prod _{j=1}^{s}{\frac {i+j+t-1}{i+j-1}}}$

Percy A. MacMahon obtained that the total number of plane partitions in ${\displaystyle {\mathcal {B}}(r,s,t)}$ is given by ${\displaystyle N_{1}(r,s,t)}$.[8] The planar case (when t = 1) yields the binomial coefficients:

${\displaystyle {\mathcal {B}}(r,s,1)={\binom {r+s}{r}}}$

## Ferrers diagrams for plane partitions

Another representation for plane partitions is in the form of Ferrers diagrams. The Ferrers diagram of a plane partition of ${\displaystyle n}$ is a collection of ${\displaystyle n}$ points or nodes, ${\displaystyle \lambda =(\mathbf {y} _{1},\mathbf {y} _{2},\ldots ,\mathbf {y} _{n})}$, with ${\displaystyle \mathbf {y} _{i}\in \mathbb {Z} _{\geq 0}^{3}}$ satisfying the condition:[9]

Condition FD: If the node ${\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3})\in \lambda }$, then so do all the nodes ${\displaystyle \mathbf {y} =(y_{1},y_{2},y_{3})}$ with ${\displaystyle 0\leq y_{i}\leq a_{i}}$ for all ${\displaystyle i=1,2,3}$.

Replacing every node of a plane partition by a unit cube with edges aligned with the axes leads to the stack of cubes representation for the plane partition.

### Equivalence of the two representations

Given a Ferrers diagram, one constructs the plane partition (as in the main definition) as follows.

Let ${\displaystyle \pi _{i,j}}$ be the number of nodes in the Ferrers diagram with coordinates of the form ${\displaystyle (i-1,j-1,*)}$ where ${\displaystyle *}$ denotes an arbitrary value. The collection ${\displaystyle \pi _{i,j}}$ form a plane partition. One can verify that condition FD implies that the conditions for a plane partition are satisfied.

Given a set of ${\displaystyle \pi _{i,j}}$ that form a plane partition, one obtains the corresponding Ferrers diagram as follows.

Start with the Ferrers diagram with no nodes. For every non-zero ${\displaystyle \pi _{i,j}}$, add ${\displaystyle \pi _{i,j}}$ nodes of the form ${\displaystyle (i-1,j-1,y_{3})}$ for ${\displaystyle 0\leq y_{3}<\pi _{i,j}-1}$ to the Ferrers diagram. By construction, it is easy to see that condition FD is satisfied.

For instance, below are shown the two representations of a plane partitions of 5.

${\displaystyle \left({\begin{smallmatrix}0\\0\\0\end{smallmatrix}}{\begin{smallmatrix}0\\0\\1\end{smallmatrix}}{\begin{smallmatrix}0\\1\\0\end{smallmatrix}}{\begin{smallmatrix}1\\0\\0\end{smallmatrix}}{\begin{smallmatrix}1\\1\\0\end{smallmatrix}}\right)\qquad \Longleftrightarrow \qquad {\begin{matrix}2&1\\1&1\end{matrix}}}$

Above, every node of the Ferrers diagram is written as a column and we have only written only the non-vanishing ${\displaystyle \pi _{i,j}}$ as is conventional.

## Action of S2, S3 and C3 on plane partitions

${\displaystyle {\mathcal {S}}_{2}}$ is the group of permutations acting on the first two coordinates of (i,j,k). This group contains the identity (i,j,k) 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 and ht(2,1,1)=2. ht(${\displaystyle \eta }$) is the height of an orbit, which is the height of any element in the orbit. This notation of the height differs from the notation of Ian G. Macdonald.[10]

There is a natural action of the permutation group ${\displaystyle {\mathcal {S}}_{3}}$ on a Ferrers diagram—this corresponds to simultaneously permuting the three coordinates of all nodes. This generalizes the conjugation operation for 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.

${\displaystyle {\begin{smallmatrix}3&1\end{smallmatrix}}\quad {\begin{smallmatrix}3\\1\end{smallmatrix}}\quad {\begin{smallmatrix}2&1&1\end{smallmatrix}}\quad {\begin{smallmatrix}2\\1\\1\end{smallmatrix}}\quad {\begin{smallmatrix}1&1&1\\1\end{smallmatrix}}\quad {\begin{smallmatrix}1&1\\1\\1\end{smallmatrix}}}$

${\displaystyle {\mathcal {C}}_{3}}$ is called the group of cyclic permutations and consists of

${\displaystyle (i,j,k)\rightarrow (i,j,k),\quad (i,j,k)\rightarrow (j,k,i),\quad {\textrm {and}}\quad (i,j,k)\rightarrow (k,i,j).}$

## Symmetric plane partitions

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

A symmetric plane partition with sum 35

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

In 1898, Percy A. 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 The MacMahon conjecture. The generating function is given by

${\displaystyle \sum _{\pi \in {\mathcal {B}}(r,r,t)/{\mathcal {S}}_{2}}q^{|\pi |}=\prod _{i=1}^{r}\left[{\frac {1-q^{t+2i-1}}{1-q^{2i-1}}}\prod _{j=i+1}^{r}{\frac {1-q^{2(i+j+t-1)}}{1-q^{2(i+j-1)}}}\right]}$

Ian G. 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 and Donald E. Knuth[12] conjectured a simple closed form for the generating function for plane partition which have at most r rows and strict decrease along the rows. George Andrews[13] showed, that the conjecture of Edward A. Bender and Donald E. 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[10] [example 16–19, pp. 83–86]. When setting q = 1 yields the counting function ${\displaystyle N_{2}(r,r,t)}$ which is given by

${\displaystyle N_{2}(r,r,t)=\prod _{i=1}^{r}{\frac {2i+t-1}{2i-1}}\prod _{1\leq i

For a proof of the case q = 1 please refer to George Andrews' paper MacMahon's conjecture on symmetric plane partitions.[15]

## Cyclically symmetric plane partitions

${\displaystyle \pi }$ is called cyclically symmetric, if the i-th row of ${\displaystyle \pi }$ is conjugate to the i-th column for all i. The i-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 (i,j,k)${\displaystyle \in {\mathcal {B}}(r,s,t)}$ then (k,i,j) and (j,k,i) are as well in ${\displaystyle {\mathcal {B}}(r,s,t)}$. Below an example of a cyclically symmetric plane partition and it's visualization is given.

A cyclically symmetric plane partition

${\displaystyle {\begin{matrix}6&5&5&4&3&3\\6&4&3&3&1&\\6&4&3&1&1&\\4&2&2&1&&\\3&1&1&&&\\1&1&1&&&\end{matrix}}}$

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

${\displaystyle \sum _{\pi \in {\mathcal {B}}(r,r,t)/{\mathcal {C}}_{3}}q^{|\pi |}=\prod _{\eta \in {\mathcal {B}}(r,r,t)/{\mathcal {C}}_{3}}{\frac {1-q^{|\eta |(1+ht(\eta ))}}{1-q^{|\eta |ht(\eta )}}}}$

This equation can also be written in another way

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

In 1979 George Andrews has proven Macdonald's conjecture for the case q=1 as the "weak" Macdonald conjecture.[16] Three years later William. H. Mills, David Robbins and Howard Rumsey proved the general case of Macdonald's conjecture in their paper Proof of the Macdonald conjecture.[17] The formula for ${\displaystyle N_{3}(r,r,r)}$ is given by the "weak" Macdonald conjecture

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

## Totally symmetric plane partitions

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. So therefore if (i,j,k)${\displaystyle \in {\mathcal {B}}(r,s,t)}$ 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.

A totally symmetric plane partition

${\displaystyle {\begin{matrix}5&4&4&3&1\\4&3&3&1&\\4&3&2&1&\\3&1&1&&\\1&&&&\end{matrix}}}$

Ian G. 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 1995 John R. Stembridge first proved the formula for ${\displaystyle N_{4}(r,r,r)}$[18] and later in 2005 it was proven by George Andrews, Peter Paule, and Carsten Schneider.[19] Around 1983 George Andrews and David Robbins independently stated an explicit product formula for the orbit-counting generating function for totally symmetric plane partitions.[20][21] This formula already alluded to in George E. Andrews' paper Totally symmetric plane partitions which was published 1980.[22] The conjecture is called The q-TSPP conjecture 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

${\displaystyle \sum _{\pi \in {\mathcal {B}}(r,r,r)/{\mathcal {S}}_{3}}q^{|\pi |}=\prod _{\eta \in {\mathcal {B}}(r,r,r)/{\mathcal {S}}_{3}}{\frac {1-q^{1+ht(\eta )}}{1-q^{ht(\eta )}}}=\prod _{1\leq i\leq j\leq k\leq r}{\frac {1-q^{i+j+k-1}}{1-q^{i+j+k-2}}}}$

This conjecture turned into a Theorem in 2011. For a proof of the q-TSPP conjecture please refer to the paper A proof of George Andrews' and David Robbins' q-TSPP conjecture by Christoph Koutschan, Manuel Kauers and Doron Zeilberger.[23]

## Self-complementary plane partitions

If ${\displaystyle \pi _{i,j}+\pi _{r-i+1,s-j+1}=t}$ for all ${\displaystyle 1\leq i\leq r}$, ${\displaystyle 1\leq j\leq s}$, 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 it's visualisation is given.

A self-complementary plane partition

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

Richard P. Stanley[24] conjectured formulas for the total number of self-complementary plane partitions ${\displaystyle N_{5}(r,s,t)}$. According to Richard Stanley, David 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 of r,s and t is even. A proof can be found in the paper Symmetries of Plane Partitions which was written by Richard P. Stanley.[25][24] The proof works with Schur functions ${\displaystyle s_{s^{r}}(x)}$. Stanley's proof of the ordinary enumeration of self-complementary plane partitions yields the q-analogue by substitutting ${\displaystyle x_{i}=q^{i}}$ for ${\displaystyle i=1,\ldots ,n}$.[26] This is a special case of Stanley's hook-content formula.[27] The generating function for self-complementary plane partitions is given by

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

Substituting this formula in

${\displaystyle s_{s^{r}}(x_{1},x_{2},\ldots ,x_{t+r})^{2}\ {\textrm {for}}\ {\mathcal {B}}(2r,2s,2t)}$

${\displaystyle s_{s^{r}}(x_{1},x_{2},\ldots ,x_{t+r})s_{(s+1)^{r}}(x_{1},x_{2},\ldots ,x_{t+r})\ {\textrm {for}}\ {\mathcal {B}}(2r,2s+1,2t)}$

${\displaystyle s_{s^{r+1}}(x_{1},x_{2},\ldots ,x_{t+r+1})s_{s^{r}}(x_{1},x_{2},\ldots ,x_{t+r+1})\ {\textrm {for}}\ {\mathcal {B}}(2r+1,2s,2t+1)}$

supplies the desired q-analogue case.

## Cyclically symmetric self-complementary plane partitions

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

A cyclically symmetric self-complementary plane partition

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

In a private communication with Richard P. Stanley, David Robbins conjectured that the total number of cyclically symmetric self-complementary plane partitions is given by ${\displaystyle N_{6}(2r,2r,2r)}$.[21][24] 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.[28]

## Totally symmetric self-complementary plane partitions

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

A totally symmetric self-complementary plane partition

${\displaystyle {\begin{matrix}6&6&6&5&5&3\\6&5&5&3&3&1\\6&5&5&3&3&1\\5&3&3&1&1&\\5&3&3&1&1&\\3&1&1&&&\end{matrix}}}$

The formula ${\displaystyle N_{7}(r,r,r)}$ was conjectured by William H. Mills, David Robbins and Howard Rumsey in their work Self-Complementary Totally Symmetric Plane Partitions.[29] The total number of totally symmetric self-complementary plane partitions is given by

${\displaystyle N_{7}(2r,2r,2r)=D_{r}}$

George Andrews has proven this formula in 1994 in his paper Plane Partitions V: The TSSCPP Conjecture.[30]

## References

1. ^ Richard P. Stanley, Enumerative Combinatorics, Volume 2. Corollary 7.20.3.
2. ^ R.P. Stanley, Enumerative Combinatorics, Volume 2. pp. 365, 401–2.
3. ^ E. M. Wright, Asymptotic partition formulae I. Plane partitions, The Quarterly Journal of Mathematics 1 (1931) 177–189.
4. ^ L. Mutafchiev and E. Kamenov, "Asymptotic formula for the number of plane partitions of positive integers", Comptus Rendus-Academie Bulgare Des Sciences 59 (2006), no. 4, 361.
5. ^ MacMahon, Percy A. (1896). "XVI. Memoir on the theory of the partition of numbers.-Part I". Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. 187: Article 52.
6. ^ MacMahon, Major Percy A. (1916). Combinatory Analysis Vol 2. Chambridge: at the University Press. pp. §495.
7. ^ MacMahon, Major Percy A. (1916). "Combinatory Analysis". Chambridge: At the University Press. 2: §429.
8. ^ MacMahon, Major Percy A. (1916). Combinatory Analysis. Chambridge: at the University Press. pp. §429, §494.
9. ^ Atkin, A. O. L.; Bratley, P.; Macdonald, I. G.; McKay, J. K. S. (1967). "Some computations for m-dimensional partitions". Proc. Camb. Phil. Soc. 63 (4): 1097–1100. Bibcode:1967PCPS...63.1097A. doi:10.1017/s0305004100042171.
10. ^ a b c d Macdonald, Ian G. (1998). Symmetric Functions and Hall Polynomials. Clarendon Press. pp. 20f, 85f. ISBN 9780198504504.
11. ^ MacMahon, Percy Alexander (1899). "Partitions of numbers whose graphs possess symmetry". Transactions of the Cambridge Philosophical Society. 17.
12. ^ Bender and Knuth (1972). "Enumeration of plane partitions". Journal of Combinatorial Theory, Series A. 13: 40–54. doi:10.1016/0097-3165(72)90007-6.
13. ^ Andrews, George E. (1977). "Plane partitions II: The equivalence of the Bender-Knuth and MacMahon conjectures". Pacific Journal of Mathematics. 72 (2): 283–291. doi:10.2140/pjm.1977.72.283.
14. ^ Andrews, George (1975). "Plane Partitions (I): The Mac Mahon Conjecture". Adv. Math. Suppl. Stud. 1.
15. ^ Andrews, George E. (1977). "MacMahon's conjecture on symmetric plane partitions". Proceedings of the National Academy of Sciences. 74 (2): 426–429. Bibcode:1977PNAS...74..426A. doi:10.1073/pnas.74.2.426. PMC 392301. PMID 16592386.
16. ^ Andrews, George E. (1979). "Plane Partitions(III): The Weak Macdonald Conjecture". Inventiones Mathematicae. 53 (3): 193–225. Bibcode:1979InMat..53..193A. doi:10.1007/bf01389763.
17. ^ Mills, Robbins, Rumsey (1982). "Proof of the Macdonald conjecture". Inventiones Mathematicae. 66: 73–88. Bibcode:1982InMat..66...73M. doi:10.1007/bf01404757.CS1 maint: Multiple names: authors list (link)
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19. ^ Andrews, Paule, Schneider (2005). "Plane Partitions VI: Stembridge's TSPP theorem". Advances in Applied Mathematics. 34 (4): 709–739. doi:10.1016/j.aam.2004.07.008.CS1 maint: Multiple names: authors list (link)
20. ^ Bressoud, David M. (1999). Proofs and Confirmations. Cambridge University Press. pp. conj. 13. ISBN 9781316582756.
21. ^ a b Stanley, Richard P. (1970). "A Baker's dozen of conjectures concerning plane partitions". Combinatoire énumérative: 285–293.
22. ^ Andrews, George (1980). "Totally symmetric plane partitions". Abstracts Amer. Math. Soc. 1: 415.
23. ^ Koutschan, Kauers, Zeilberger (2011). "A proof of George Andrews' and David Robbins' q-TSPP conjecture". PNAS. 108 (6): 2196–2199. arXiv:1002.4384. Bibcode:2011PNAS..108.2196K. doi:10.1073/pnas.1019186108.CS1 maint: Multiple names: authors list (link)
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25. ^ "Erratum". Journal of Combinatorial Theory. 43: 310. 1986.
26. ^ Eisenkölbl, Theresia (2008). "A Schur function identity related to the (-1)-enumeration of self complementary plane partitions". Journal of Combinatorial Theory, Series A. 115: 199–212. doi:10.1016/j.jcta.2007.05.004.
27. ^ Stanley, Richard P. (1971). "Theory and Application of Plane Partitions. Part 2". Advances in Applied Mathematics. 50 (3): 259–279. doi:10.1002/sapm1971503259.
28. ^ Kuperberg, Greg (1994). "Symmetries of plane partitions and the permanent-determinant method". Journal of Combinatorial Theory, Series A. 68: 115–151. arXiv:math/9410224. Bibcode:1994math.....10224K. doi:10.1016/0097-3165(94)90094-9.
29. ^ Mills, Robbins, Rumsey (1986). "Self-Complementary Totally Symmetric Plane Partitions". Journal of Combinatorial Theory, Series A. 42 (2): 277–292. doi:10.1016/0097-3165(86)90098-1.CS1 maint: Multiple names: authors list (link)
30. ^ Andrews, George E. (1994). "Plane Partitions V: The TSSCPP Conjecture". Journal of Combinatorial Theory, Series A. 66: 28–39. doi:10.1016/0097-3165(94)90048-5.