A plane partition of a nonnegative integer is a finite two-dimensional array of
positive integers in the shape of a Ferrers
diagram, with entries nonincreasing both from left to right and top to bottom
and summing to .
The empty array is the unique plane partition of 0. In other words, whenever the
indicated entries exist,
(1)
(2)
and
(3)
The Ferrers diagram shape means that the rows are aligned at the left, their lengths are nonincreasing from top to bottom, and
there are no holes.
(4)
For example, one plane partition of 22 is illustrated above.
MacMahon (1960) also showed that the number of plane partitions whose Ferrers
diagrams fit inside an rectangle and whose entries
do not exceed
is given by the following formula, where , , and are nonnegative integers,
and the empty plane partition is included,
(8)
(Bressoud and Propp 1999, Fulmek and Krattenthaler 2000). Expanding out the products 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).
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, 1999. https://www.ams.org/notices/199906/fea-bressoud.pdf.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. Monthly96,
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. A211, 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. A211, 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.
https://www.inwap.com/pdp10/hbaker/hakmem/boolean.html#item18.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). Berlin, Germany: Springer-Verlag, pp. 285-293,
1986a. https://doi.org/10.1007/BFb0072521.Stanley,
R. P. "Symmetries of Plane Partitions." J. Combin. Theory Ser.
A43, 103-113, 1986b. https://doi.org/10.1016/0097-3165(86)90028-2.
is a finite two-dimensional array of
positive integers 
in the shape of a Ferrers
diagram, with entries nonincreasing both from left to right and top to bottom
and summing to 
.
The empty array is the unique plane partition of 0. In other words, whenever the
indicated entries exist,
of plane partitions of 
is
,
a recurrence equation for 
with 
and 
is given by
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.svg)
of plane partitions whose Ferrers
diagrams fit inside an 
rectangle and whose entries
do not exceed 
is given by the following formula, where 
, 
, and 
are nonnegative integers,
and the empty plane partition is included,
is the Barnes G-function. Taking 
gives
![product_(i=1)^(n)(Gamma(i)Gamma(i+2n))/([Gamma(i+n)]^2)](/images/equations/PlanePartition/Inline27.svg)
![([G(n+1)]^3G(3n+1))/([G(2n+1)]^3),](/images/equations/PlanePartition/Inline30.svg)
also gives the number of hexagon
tilings by rhombi for a hexagon
of side lengths 
,
, 
, 
, 
, 
(David and Tomei 1989, Fulmek and Krattenthaler 2000).
