A plane partition is a two-dimensional array of integers that are nonincreasing both from left to right and top to bottom and that add up to a given number . In other words,

(1)

(2)

and

(3)

Implicit in this definition is the requirement that the array be flush on top and to the left and contain no holes.

(4)

For example, one plane partition of 22 is illustrated above.

The generating function for the number of planar partitions of is

(5)

(Sloane's A000219, MacMahon 1912b, Speciner 1972, Bender and Knuth 1972, Bressoud and Propp 1999).

MacMahon (1960) also showed that the number of plane partitions whose Young diagrams fit inside an rectangle and whose integers do not exceed (in other words, with all ) is given by

(6)

(Bressoud and Propp 1999, Fulmek and Krattenthaler 2000). Expanding out the products gives

			(7)
			(8)
			(9)

where is the Barnes G-function. Taking gives

			(10)
			(11)

the first few terms of which are 2, 20, 980, 232848, 267227532, 1478619421136, ... (Sloane's A008793). Amazingly, also gives the number of hexagon tilings by rhombi for a hexagon of side lengths , , , , , (David and Tomei 1989, Fulmek and Krattenthaler 2000).

The concept of planar partitions can also be generalized to cubic partitions.

Bender, E. A. and Knuth, D. E. "Enumeration of Plane Partitions." J. Combin. Theory Ser. A. 13, 40-54, 1972.

Bressoud, D. Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture. Cambridge, England: Cambridge University Press, 1999.

Bressoud, D. and Propp, J. "How the Alternating Sign Matrix Conjecture was Solved." Not. Amer. Math. Soc. 46, 637-646.

Cohn, H.; Larsen, M.; and Propp, J. "The Shape of a Typical Boxed Plane Partition." New York J. Math. 4, 137-166, 1998.

David, G. and Tomei, C. "The Problem of the Calissons." Amer. Math. Monthly 96, 429-431, 1989.

Fulmek, M. and Krattenthaler, C. "The Number of Rhombus Tilings of a Symmetric Hexagon which Contains a Fixed Rhombus on the Symmetry Axes, II." Europ. J. Combin. 21, 601-640, 2000.

Knuth, D. E. "A Note on Solid Partitions." Math. Comput. 24, 955-961, 1970.

MacMahon, P. A. "Memoir on the Theory of the Partitions of Numbers. V: Partitions in Two-Dimensional Space." Phil. Trans. Roy. Soc. London Ser. A 211, 75-110, 1912a.

MacMahon, P. A. "Memoir on the Theory of the Partitions of Numbers. VI: Partitions in Two-Dimensional Space, to which is Added an Adumbration of the Theory of Partitions in Three-Dimensional Space." Phil. Trans. Roy. Soc. London Ser. A 211, 345-373, 1912b.

MacMahon, P. A. §429 and 494 in Combinatory Analysis, Vol. 2. New York: Chelsea, 1960.

Mills, W. H.; Robbins, D. P.; and Rumsey, H. Jr. "Proof of the Macdonald Conjecture." Invent. Math. 66, 73-87, 1982.

Sloane, N. J. A. Sequences A000219/M2566 and A008793 in "The On-Line Encyclopedia of Integer Sequences."

Speciner, M. Item 18 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 10, Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/boolean.html#item18.

Stanley, R. P. "Symmetry of Plane Partitions." J. Combin. Th. Ser. A 3, 103-113, 1986.

Stanley, R. P. "A Baker's Dozen of Conjectures Concerning Plane Partitions." In Combinatoire Énumérative: Proceedings of the "Colloque De Combinatoire Enumerative," Held at Université Du Quebec a Montreal, May 28-June 1, 1985 (Ed. G. Labelle and P. Leroux). New York: Springer-Verlag, 285-293, 1986.

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Weisstein, Eric W. "Plane Partition." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PlanePartition.html