Plane Partition

A two-dimensional array of Integers nonincreasing both left to right and top to bottom which add up to a given number, i.e., $n_{ij}\geq n_{i(j+1)}$ and $n_{ij}\geq n_{(i+1)j}$. For example, a planar partition of 22 is given by

$$\matrix{ 5 & 4 & 2 & 1 & 1\cr 3 & 2\cr 2 & 2.\cr}$$

The Generating Function for the number ${\rm PL}(n)$ of planar partitions of $n$ is

$$\sum_{n=0}^\infty \mathop{\rm PL}(n)x^n={1\over\prod_{k=1}^\infty (1-x^k)^k} =1+x+3x^2+6x^3+13x^4+24x^5+\ldots$$

(Sloane's A000219, MacMahon 1912b, Beeler et al. 1972, Bender and Knuth 1972). The concept of planar partitions can also be generalized to cubic partitions.

See also Partition, Solid Partition

References

Beeler, M.; Gosper, R. W.; and Schroeppel, R. Item 18 in HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.

Bender, E. A. and Knuth, D. E. ``Enumeration of Plane Partitions.'' J. Combin. Theory Ser. A. 13, 40-54, 1972.

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. Combinatory Analysis, Vol. 2. New York: Chelsea, 1960.

Sloane, N. J. A. Sequence A000219/M2566 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.