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Wikipedia / Plane partition

In mathematics and especially in combinatorics, a plane partition (with positive integer indices i and j) that is nonincreasing in both indices. This means that

  1. for all i and j.

may be nonzero. Plane partitions are a generalization of partitions of an integer.

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

of positive integer lattice points (i, j, k, such that if (r, s, t, then (i, j, k.

The sum of a plane partition is

The sum describes the number of cubes of which the plane partition consists. Much interest in plane partitions concerns the enumeration of plane partitions in various classes. The number of plane partitions with sum n 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.

Generating function of plane partitions

The generating function for PL(n) is

  1. .

It is sometimes referred to as the MacMahon function, as it was discovered by Percy A. MacMahon.

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). The asymptotics for plane partitions were first calculated by E. M. Wright., that

Evaluating numerically yields

Plane partitions in a box

in his first paper on plane partitions.

A proof of this formula can be found in the book Combinatory Analysis written by MacMahon. MacMahon also mentions the generating functions of plane partitions.

, and setting q= is equal to the following product formula: The planar case (when t = 1) yields the binomial coefficients:

The general solution is

The isometric projection of the unit cubes representing a plane partition in a box gives a bijection between these plane partitions and rhombus tilings of a hexagon with the same edge lengths as the box.

Special plane partitions

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

for the number of such plane partitions, where , , and are the dimensions of the box under consideration, and is the index for the case being considered.

Action of S2, S3 and C3 on plane partitions

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 height of an element (i, j, k 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.

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 action. Only the exchange of the first two coordinates is manifest in the representation given below.

is called the group of cyclic permutations and consists of

Symmetric plane partitions

is called symmetric if i,j = j,i for all i, j. 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.

. This conjecture is called the MacMahon conjecture

Macdonald pointed out that Percy A. MacMahon's conjecture reduces to

In 1972 Edward A. Bender and Donald E. Knuth conjectured a simple closed form for the generating function for plane partitions which have at most r rows and strict decrease along the rows. George Andrews showed 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 and later Ian G. Macdonald presented an alternative proof. When setting q= which is given by

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

Cyclically symmetric plane partitions

is called cyclically symmetric, if the i is conjugate to the i-th column for all i. The i. then (k, i, j) and (j, k, i. 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 integer r. This conjecture is called the Macdonald conjecture is given by

This equation can also be written in another way

In 1979, Andrews proved Macdonald's conjecture for the case q=1 as the "weak" Macdonald conjecture. 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. is given by the "weak" Macdonald conjecture

Totally symmetric plane partitions

then all six permutations of (i, j, k. Below an example of a matrix for a totally symmetric plane partition is given. The picture shows the visualisation of the matrix.

. The formula is given by

In 1995 John R. Stembridge and later in 2005 it was proven by George Andrews, Peter Paule, and Carsten Schneider. Around 1983 Andrews and Robbins independently stated an explicit product formula for the orbit-counting generating function for totally symmetric plane partitions. This formula already alluded to in George E. Andrews' paper Totally symmetric plane partitions which was published 1980. The conjecture is called The q-TSPP conjecture and it is given by:

is given by the formula

This conjecture was proved in 2011 by Christoph Koutschan, Manuel Kauers and Doron Zeilberger.

Self-complementary plane partitions

is even. Below an example of a self-complementary symmetric plane partition and its visualisation is given.

Richard P. Stanley is given by

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.. Stanley's proof of the ordinary enumeration of self-complementary plane partitions yields the q. This is a special case of Stanley's hook-content formula. The generating function for self-complementary plane partitions is given by

Substituting this formula in

supplies the desired q-analogue case.

Cyclically symmetric self-complementary plane partitions

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.

. The total number of cyclically symmetric self-complementary plane partitions is given by

alternating sign matrices is given by

Greg Kuperberg in 1994.

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.

was conjectured by William H. Mills, Robbins and Howard Rumsey in their work Self-Complementary Totally Symmetric Plane Partitions. The total number of totally symmetric self-complementary plane partitions is given by

Andrews proved this formula in 1994 in his paper Plane Partitions V: The TSSCPP Conjecture.

See also

References