Plane Partition
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)
|
(OEIS A000219, MacMahon 1912b, Speciner 1972,
Bender and Knuth 1972, Bressoud and Propp 1999).
Writing 
, a recurrence
equation for 
is given by
 |
(6)
|
where 
is a divisor
function. It also has generating function
![G(x)=exp[sum_(n=1)^inftysigma_2(n)(x^n)/n].](/images/equations/PlanePartition/NumberedEquation7.gif) |
(7)
|
MacMahon (1960) also showed that the number 
of plane
partitions whose Young tableaux fit inside an 
rectangle and whose integers do not exceed
(in other words, with all 
) is
given by
 |
(8)
|
(Bressoud and Propp 1999, Fulmek and Krattenthaler 2000). Expanding out the products gives
where 
is the Barnes
G-function. Taking 
gives
the first few terms of which are 2, 20, 980, 232848, 267227532, 1478619421136, ... (OEIS 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.
SEE ALSO: Cyclically Symmetric Plane Partition,
Descending
Plane Partition,
Hexagon Tiling,
Partition,
Macdonald's Plane Partition Conjecture,
Solid Partition,
Totally
Symmetric Self-Complementary Plane Partition,
Young
Tableau
REFERENCES:
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.
Referenced on Wolfram|Alpha:
Plane Partition
CITE THIS AS:
Weisstein, Eric W. "Plane Partition."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PlanePartition.html
![product_(i=1)^(n)(Gamma(i)Gamma(i+2n))/([Gamma(i+n)]^2)](/images/equations/PlanePartition/Inline22.gif)
![([G(n+1)]^3G(3n+1))/([G(2n+1)]^3),](/images/equations/PlanePartition/Inline25.gif)
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