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 $\pi _{i,j}$ (with positive integer indices i and j) that is nonincreasing in both indices. This means that

\pi _{i,j}\geq \pi _{i,j+1}\quad

and

\quad \pi _{i,j}\geq \pi _{i+1,j}\,

for all i and j.

Moreover only finitely many of the $\pi _{i,j}$ are nonzero. A plane partitions may be represented visually by the placement of a stack of $\pi _{i,j}$ 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

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:

{\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 $N_{i}(r,s,t)$ 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 ${\mathcal {P}}$ of positive integer lattice points (i, j, k) in $\mathbb {N} ^{3}$ , such that if (r, s, t) lies in ${\mathcal {P}}$ and if (i, j, k) satisfies $1\leq i\leq r$ , $1\leq j\leq s$ and $1\leq k\leq t$ , then (i, j, k) also lies in ${\mathcal {P}}$ .

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

Generating function of plane partitions

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

\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 $n$ :

{\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

\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 ${\mathcal {B}}(r,s,t)$ in his first paper on plane partitions.^[5] The formula is given by

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

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

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

Setting q=1 in the formulas above yields

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

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

{\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 $n$ is a collection of $n$ points or nodes, $\lambda =({\mathbf {y}}_{1},{\mathbf {y}}_{2},\ldots ,{\mathbf {y}}_{n})$ , with ${\mathbf {y}}_{i}\in {\mathbb {Z}}_{{\geq 0}}^{{3}}$ satisfying the condition:^[9]

Condition FD: If the node

{\mathbf {a}}=(a_{1},a_{2},a_{3})\in \lambda

, then so do all the nodes

{\mathbf {y}}=(y_{1},y_{2},y_{3})

with

0\leq y_{i}\leq a_{i}

for all

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

\pi _{i,j}

be the number of nodes in the Ferrers diagram with coordinates of the form

(i-1,j-1,*)

where

*

denotes an arbitrary value. The collection

\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 $\pi _{i,j}$ 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

\pi _{i,j}

, add

\pi _{i,j}

nodes of the form

(i-1,j-1,y_{3})

for

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.

\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 $\pi _{i,j}$ as is conventional.

Action of S₂, S₃ and C₃ on plane partitions

${\mathcal {S}}_{2}$ 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 $\eta$ is denoted by | $\eta$ |. ${\mathcal {B}}/{\mathcal {S}}_{2}$ denotes the set of orbits of elements of ${\mathcal {B}}$ under the action of ${\mathcal {S}}_{2}$ . The height of an element (i,j,k) is defined by

$ht(i,j,k)=i+j+k-2$

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( $\eta$ ) 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 ${\mathcal {S}}_{3}$ 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 ${\mathcal {S}}_{3}$ 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 ${\mathcal {S}}_{3}$ action. Only the exchange of the first two coordinates is manifest in the representation given below.

{\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}}

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

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

Symmetric plane partitions

A plane partition $\pi$ is called symmetric if $\pi$ _i_,j = $\pi$ _j_,i for all i,j . In other words, a plane partition is symmetric if (i,j,k) $\in {\mathcal {B}}(r,s,t)$ if and only if (j,i,k) $\in {\mathcal {B}}(r,s,t)$ . 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

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

In 1898, Percy A. MacMahon formulated his conjecture about the generating function for symmetric plane partitions which are subsets of ${\mathcal {B}}(r,r,t)$ .^[11] This conjecture is called The MacMahon conjecture. The generating function is given by

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

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

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

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, p. 83-86]. When setting q = 1 yields the counting function $N_{2}(r,r,t)$ which is given by

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

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

$\pi$ is called cyclically symmetric, if the i-th row of $\pi$ 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 $\pi$ is the partition whose diagram is the transpose of partition $\pi$ .^[10] In other words, the plane partition is cyclically symmetric if whenever (i,j,k) $\in {\mathcal {B}}(r,s,t)$ then (k,i,j) and (j,k,i) are as well in ${\mathcal {B}}(r,s,t)$ . Below an example of a cyclically symmetric plane partition and it's visualization is given.

A cyclically symmetric plane partition

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

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 ${\mathcal {B}}(r,r,r)$ is given by

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

This equation can also be written in another way

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

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 $N_{3}(r,r,r)$ is given by the "weak" Macdonald conjecture

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

Totally symmetric plane partitions

A totally symmetric plane partition $\pi$ 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) $\in {\mathcal {B}}(r,s,t)$ then all six permutations of (i,j,k) are also in ${\mathcal {B}}(r,s,t)$ . 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

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

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

$N_{4}(r,r,r)=\prod _{\eta \in {\mathcal {B}}(r,r,r)/{\mathcal {S}}_{3}}{\frac {1+ht(\eta )}{ht(\eta )}}$

In 1995 John R. Stembridge first proved the formula for $N_{4}(r,r,r)$ ^[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 ${\mathcal {S}}_{3}$ be the symmetric group. The orbit counting function for totally symmetric plane partitions that fit inside ${\mathcal {B}}(r,r,r)$ is given by the formula

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

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 $\pi _{i,j}+\pi _{r-i+1,s-j+1}=t$ for all $1\leq i\leq r$ , $1\leq j\leq s$ , then the plane partition is called self-complementary. It is necessary that the product $r\cdot s\cdot t$ is even. Below an example of a self-complementary symmetric plane partition and it's visualisation is given.

A self-complementary plane partition

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

Richard P. Stanley^[24] conjectured formulas for the total number of self-complementary plane partitions $N_{5}(r,s,t)$ . 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 ${\mathcal {B}}(r,s,t)$ is given by

$N_{5}(2r,2s,2t)=N_{1}(r,s,t)^{2}$

$N_{5}(2r+1,2s,2t)=N_{1}(r,s,t)N_{1}(r+1,s,t)$

$N_{5}(2r+1,2s+1,2t)=N_{1}(r+1,s,t)N_{1}(r,s+1,t)$

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 $s_{s^{r}}(x)$ . Stanley's proof of the ordinary enumeration of self-complementary plane partitions yields the q-analogue by substitutting $x_{i}=q^{i}$ for $i=1,\ldots ,n$ .^[26] This is a special case of Stanley's hook-content formula.^[27] The generating function for self-complementary plane partitions is given by

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

Substituting this formula in

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

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

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

supplies the desired q-analogue case.

Cyclically symmetric self-complementary plane partitions

A plane partition $\pi$ 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

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

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 $N_{6}(2r,2r,2r)$ .^[21]^[24] The total number of cyclically symmetric self-complementary plane partitions is given by

$N_{6}(2r,2r,2r)=D_{r}^{2}$

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

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

Greg Kuperberg proved the formula for $N_{6}(r,r,r)$ 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

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

The formula $N_{7}(r,r,r)$ 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

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

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

References

↑ Richard P. Stanley, Enumerative Combinatorics, Volume 2. Corollary 7.20.3.
↑ R.P. Stanley, Enumerative Combinatorics, Volume 2. pp. 365, 401–2.
↑ E. M. Wright, Asymptotic partition formulae I. Plane partitions, The Quarterly Journal of Mathematics 1 (1931) 177–189.
↑ 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.
↑ 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.
↑ MacMahon, Major Percy A. (1916). Combinatory Analysis Vol 2. Chambridge: at the University Press. pp. §495.
↑ MacMahon, Major Percy A. (1916). "Combinatory Analysis". Chambridge: at the University Press. 2: §429.
↑ MacMahon, Major Percy A. (1916). Combinatory Analysis. Chambridge: at the University Press. pp. §429, §494.
↑ 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: 1097–1100. Bibcode:1967PCPS...63.1097A. doi:10.1017/s0305004100042171.
1 2 3 4 Macdonald, Ian G. (1998). Symmetric Functions and Hall Polynomials. Clarendon Press. pp. 20f, 85f. ISBN 9780198504504.
↑ MacMahon, Percy Alexander (1899). "Partitions of numbers whose graphs possess symmetry". Transactions of the Cambridge Philosophical Society. 17.
↑ 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.
↑ Andrews, George E. (1977). "Plane partitions II: The equivalence of the Bender-Knuth and MacMahon conjectures". Pacific Journal of Mathematics. 72: 283–291. doi:10.2140/pjm.1977.72.283.
↑ Andrews, George (1975). "Plane Partitions (I): The Mac Mahon Conjecture". Adv. Math. Suppl. Stud. 1.
↑ Andrews, George E. (1977). "MacMahon's conjecture on symmetric plane partitions". Proceedings of the National Academy of Sciences. 74: 426–429. Bibcode:1977PNAS...74..426A. doi:10.1073/pnas.74.2.426. PMC 392301.
↑ Andrews, George E. (1979). "Plane Partitions(III): The Weak Macdonald Conjecture". Inventiones Mathematicae. 53: 193–225. Bibcode:1979InMat..53..193A. doi:10.1007/bf01389763.
↑ Mills, Robbins, Rumsey (1982). "Proof of the Macdonald conjecture". Inventiones Mathematicae. 66: 73–88. Bibcode:1982InMat..66...73M. doi:10.1007/bf01404757.
↑ Stembridge, John R. (1995). "The Enumeration of Totally Symmetric Plane Partitions". Advances in Mathematics. 111: 227–243. doi:10.1006/aima.1995.1023.
↑ Andrews, Paule, Schneider (2005). "Plane Partitions VI: Stembridge's TSPP theorem". Advances in Applied Mathematics. 34: 709–739. doi:10.1016/j.aam.2004.07.008.
↑ Bressoud, David M. (1999). Proofs and Confirmations. Cambridge University Press. pp. conj. 13. ISBN 9781316582756.
1 2 Stanley, Richard P. (1970). "A baker's dozon of conjectures concering plane partitions". Combinatoire énumérative: 285–293.
↑ Andrews, George (1980). "Totally symmetric plane partitions". Abstracts Amer. Math. Soc. 1: 415.
↑ Koutschan, Kauers, Zeilberger (2011). "A proof of George Andrews' and David Robbins' q-TSPP conjecture". PNAS. 108: 2196–2199. arXiv:1002.4384. Bibcode:2011PNAS..108.2196K. doi:10.1073/pnas.1019186108.
1 2 3 Stanley, Richard P. (1986). "Symmetries of Plane Partitions". Journal of Combinatorial Theory, Series A. 43: 103–113. doi:10.1016/0097-3165(86)90028-2.
↑ "Erratum". Journal of Combinatorial Theory. 43: 310. 1986.
↑ 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.
↑ Stanley, Richard P. (1971). "Theory and Application of Plane Partitions. Part 2". Advances in Applied Mathematics. 50: 259–279. doi:10.1002/sapm1971503259.
↑ Kuperberg, Greg (1994). "Symmetries of plane partitions and the permanent-determinant method". Journal of Combinatorial Theory, Series A. 68: 115–151. doi:10.1016/0097-3165(94)90094-9.
↑ Mills, Robbins, Rumsey (1986). "Self-Complementary Totally Symmetric Plane Partitions". Journal of Combinatorial Theory, Series A. 42: 277–292. doi:10.1016/0097-3165(86)90098-1.
↑ 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.

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.

External links

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[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.

[:5-5] 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.

[:4-6] 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.

[Atkin1967-9] 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: 1097–1100. Bibcode:1967PCPS...63.1097A. doi:10.1017/s0305004100042171.

[:0-10] 1 2 3 4 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.

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