page_title: Plane partition
section_title: Cyclically symmetric self-complementary plane partitions
hierarchical_section_title: Plane partition / Cyclically symmetric self-complementary plane partitions
caption_attribution_description: English: Example of a cyclically symmetric self-complementary 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 plane partition  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.
  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 . The total number of cyclically symmetric self-complementary plane partitions is given by
   is the number of  alternating sign matrices. A formula for  is given by
  Greg Kuperberg proved the formula for  in 1994.
caption: A cyclically symmetric self-complementary plane partition
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