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In mathematics and especially in combinatorics, a plane partition is a two-dimensional array of nonnegative integers π i , j {\displaystyle \pi _{i,j}} that is nonincreasing in both indices. This means that
Moreover, only finitely many of the π i , j {\displaystyle \pi _{i,j}} may be nonzero. Plane partitions are a generalization of partitions of an integer.
A plane partition may be represented visually by the placement of a stack of π i , j {\displaystyle \pi _{i,j}} unit cubes above the point 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 P {\displaystyle {\mathcal {P}}} of positive integer lattice points in N 3 {\displaystyle \mathbb {N} ^{3}} , such that if lies in P {\displaystyle {\mathcal {P}}} and if {\displaystyle } satisfies 1 ≤ i ≤ r {\displaystyle 1\leq i\leq r} , 1 ≤ j ≤ s {\displaystyle 1\leq j\leq s} , and 1 ≤ k ≤ t {\displaystyle 1\leq k\leq t} , then also lies in P {\displaystyle {\mathcal {P}}}.