Plane partition

In mathematics and especially in combinatorics, a plane partition is a two-dimensional array of nonnegative integers ${\displaystyle n_{i,j}}$ (with positive integer indices i and j) that is nonincreasing in both indices, that is, that satisfies

and for which only finitely many of the n_i,j are nonzero. A plane partitions may be represented visually by the placement of a stack of ${\displaystyle n_{i,j}}$ unit cubes above the point (i,j) in the plane, giving a three-dimensional solid like the one shown at right.

The sum of a plane partition is

and PL(n) denotes the number of plane partitions with sum n.

For example, there are six plane partitions with sum 3:

so PL(3) = 6. (Here the plane partitions are drawn using matrix indexing for the coordinates and the entries equal to 0 are suppressed for readability.)

Another representation for plane partitions is in the form of Ferrers diagrams. The Ferrers diagram of a plane partition of ${\displaystyle n}$ is a collection of ${\displaystyle n}$ points or nodes, ${\displaystyle \lambda =(\mathbf {y} _{1},\mathbf {y} _{2},\ldots ,\mathbf {y} _{n})}$, with ${\displaystyle \mathbf {y} _{i}\in \mathbb {Z} _{\geq 0}^{3}}$ satisfying the condition:

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