Plane partition

A plane partition (parts as heights)

In mathematics, a plane partition (see also solid partition for three-dimensional array) is a two-dimensional array of nonnegative integers which are nonincreasing from left to right and top to bottom:

Thinking of the stack of unit cubes placed on (i,j)-square, we obtain a solid (or 3-dimensional) partition.

Define the sum of the plane partition by

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

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

so PL(3) = 6.

Generating function

By a result of Percy MacMahon, the generating function for PL(n) can be calculated by

This is usually referred to as the MacMahon function. This formula is the 2-dimensional analogue of Euler's product formula for the number of integer partitions of n. There is no analogous formula for partitions in higher dimensions.^{[citation needed]}

MacMahon formula

Denote by the number of solid partitions which fit into box. In the planar case, we obtain the binomial coefficients:

MacMahon formula is the multiplicative formula for general values of :

This formula was obtained by Percy MacMahon and was later rewritten in this form by Ian Macdonald.

References

• G. Andrews, The Theory of Partitions, Cambridge University Press, Cambridge, 1998, ISBN 0-521-63766-X
• Bender, Edward A.; Knuth, Donald E. (1972), "Enumeration of plane partitions", Journal of Combinatorial Theory. Series A 13: 40–54, doi:10.1016/0097-3165(72)90007-6, ISSN 1096-0899, MR0299574 
• I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, 1999, ISBN 0-19-850450-0
• P.A. MacMahon, Combinatory analysis, 2 vols, Cambridge University Press, 1915-16.

External links

• Weisstein, Eric W., "Plane partition" from MathWorld.
• (sequence A000219 in OEIS).