page_title: Plane partition
section_title: Totally symmetric plane partitions
hierarchical_section_title: Plane partition / Totally symmetric plane partitions
caption_attribution_description: English: Example of a totally symmetric plane partition
context_page_description: In mathematics and especially in combinatorics, a plane partition is a two-dimensional array of nonnegative integers that is nonincreasing in both indices. This means that
  and for all i and j.
  Moreover only finitely many of the are nonzero. A plane partitions may be represented visually by the placement of a stack of unit cubes above the point in the plane, giving a three-dimensional solid as shown in the picture.
  The sum of a plane partition is
  The sum describes the number of cubes of which the plane partition consists. The number of plane partitions with sum n is denoted PL.
  For example, there are six plane partitions with sum 3:
  so PL = 6. Let 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 of positive integer lattice points in, such that if lies in and if satisfies, and, then also lies in.
context_section_description: A totally symmetric plane partition  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) then all six permutations of (i,j,k) are also in . Below an example of a matrix for a totally symmetric plane partition is given. The picture shows the visualisation of the matrix.
  Ian G. Macdonald found the total number of totally symmetric plane partitions that are subsets of . The formula is given by
  In 1995 John R. Stembridge first proved the formula for  and later in 2005 it was proven by George Andrews, Peter Paule, and Carsten Schneider. Around 1983 George Andrews and David 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:
  Let  be the symmetric group. The orbit counting function for totally symmetric plane partitions that fit inside  is given by the formula
  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.
caption: A totally symmetric plane partition
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