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Plane partitions
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 :
Contents
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
- Partitions (one-dimensional partitions)
- Plane partitions (two-dimensional partitions)
- Solid partitions (three-dimensional partitions)
- Compositions (one-dimensional ordered partitions)
- Plane compositions (two-dimensional ordered partitions)
- Solid compositions (three-dimensional ordered partitions)
- Prime signature (which may be viewed as a given partition of , the number of prime factors (with repetition) of .)
External links
- Donald E. Knuth, A Note on Solid Partitions, 1970.
- E. Maitland Wright, ASYMPTOTIC PARTITION FORMULAE, Q J Math (1931) os-2 (1): 177-189.
- Ljuben R. Mutafchiev, Asymptotic Enumeration of Plane Partitions of Large Integers and Hayman’s Theorem for Admissible Generating Functions, American University in Bulgaria.
- Weisstein, Eric W., Plane Partition, from MathWorld—A Wolfram Web Resource..