Plane partitions in a box

Around 1896, MacMahon set up the generating function of plane partitions that are subsets of the ${\displaystyle r\times s\times t}$ box ${\displaystyle {\mathcal {B}}(r,s,t)=\{(i,j,k)|1\leq i\leq r,1\leq j\leq s,1\leq k\leq t\}}$ 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 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 ${\displaystyle \textstyle {\frac {1-q}{1-q}}}$, and setting q = 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 (when t = 1) yields the binomial coefficients:

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

The general solution is

${\displaystyle {\begin{aligned}{\mathcal {B}}(r,s,t)&=\prod _{k=1}^{t}{\frac {(r+s+k-1)!(k-1)!}{(r+k-1)!(s+k-1)!}}\\&=\prod _{k=1}^{t}{\frac {{\binom {r+s+k-1}{r+k-1}}{\binom {r+s+k-1}{s+k-1}}}{{\binom {r+s+k-1}{r}}{\binom {s+k-1}{s}}}}\end{aligned}}}$

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 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 and ht(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 of Ian G. Macdonald.^[9]

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 for integer 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

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

Symmetric plane partitions

A plane partition ${\displaystyle \pi }$ is called symmetric if π_i,j = π_j,i for all i, 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 plane x = y. Below is an example of a symmetric plane partition and its visualisation.

A symmetric plane partition, 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, MacMahon formulated his conjecture about the generating function for symmetric plane partitions which are subsets of ${\displaystyle {\mathcal {B}}(r,r,t)}$.^[10] This conjecture is called The MacMahon conjecture. The generating function is given by

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

Macdonald^[9] 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 conjectured^[11] 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 showed^[12] 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^[13] and later Ian G. Macdonald presented an alternative proof.^[14] When setting q = 1 yields the counting function ${\displaystyle N_{2}(r,r,t)}$ which is given by

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

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

π 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 ${\displaystyle \pi }$.^[9] 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.

A cyclically symmetric plane partition

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

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

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

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 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, 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.

A totally symmetric plane partition

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

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 Andrews and 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 was proved in 2011 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 its 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 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 of r,s and t is even. A proof can be found in the paper Symmetries of Plane Partitions which was written by 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 substituting ${\displaystyle x_{i}=q^{i}}$ 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

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

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

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

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 Stanley, 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, 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

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

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