Plane partition
In mathematics and especially in combinatorics, a plane partition is a two-dimensional array of nonnegative integers
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
Many symmetric classes of plane partitions are enumerated by simple product formulas.
, that[a] Evaluating numerically yields Around 1896, MacMahon set up the generating function of plane partitions that are subsets of the
A proof of this formula can be found in the book Combinatory Analysis written by MacMahon.
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.
In the subsequent sections, the enumeration of special sub-classes of plane partitions inside a box are considered.
for the number of such plane partitions, where r, s, and t are the dimensions of the box under consideration, and i is the index for the case being considered.
on a Ferrers diagram of a plane partition—this corresponds to simultaneously permuting the three coordinates of all nodes.
In 1898, MacMahon formulated his conjecture about the generating function for symmetric plane partitions which are subsets of
Macdonald[10] pointed out that Percy A. MacMahon's conjecture reduces to In 1972 Edward A. Bender and Donald E. Knuth conjectured[12] a simple closed form for the generating function for plane partition which have at most r rows and strict decrease along the rows.
MacMahon's conjecture was proven almost simultaneously by George Andrews in 1977[14] and later Ian G. Macdonald presented an alternative proof.
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.
The generating function for cyclically symmetric plane partitions which are subsets of
[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' paper Totally symmetric plane partitions which was published 1980.
The orbit counting function for totally symmetric plane partitions that fit inside
is given by the formula This conjecture was proved in 2011 by Christoph Koutschan, Manuel Kauers and Doron Zeilberger.
Richard P. Stanley[25] conjectured formulas for the total number of self-complementary plane partitions
According to Stanley, Robbins also formulated formulas for the total number of self-complementary plane partitions in a different but equivalent form.
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-analogue by substituting
[28] The generating function for self-complementary plane partitions is given by Substituting this formula in supplies the desired q-analogue case.
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
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.
