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Plane partitions

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An ordinary (i.e. one-dimensional) partition is a row of nonnegative integers in nonincreasing (weakly decreasing) order whose sum is .

A plane partition of is a two-dimensional arrangement of nonnegative integers

which are nonincreasing (weakly decreasing) in both rows and columns:

and which sum to :

Number of plane partitions

The number of planar partitions of gives the sequence (Cf. A000219)

{1, 1, 3, 6, 13, 24, 48, 86, 160, 282, 500, 859, 1479, 2485, 4167, 6879, 11297, 18334, 29601, 47330, 75278, 118794, 186475, 290783, 451194, 696033, 1068745, 1632658, 2483234, 3759612, 5668963, ...}

MacMahon's formula

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

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

Recurrence

where is the sum of squares of divisors of (cf. divisor function.)

Generating function

The generating function of ordinary (one-dimensional) partitions is the reciprocal of Euler's function

In 1912, Major Percy A. MacMahon proved that the generating function for plane partitions is

Now, you may be tempted to conjecture the generating function for solid partitions...

Also

where is the sum of squares of divisors of (cf. divisor function.)

Asymptotic behaviour

...

See also




External links