Plane partition

In mathematics, a plane partition (also solid partition) is a two-dimensional array of nonnegative integers $n\_\{i,j\}$ which are nonincreasing from left to right and top to bottom::$n\_\{i,j\}\; ge\; n\_\{i,j+1\}\; quadmbox\{and\}quad\; n\_\{i,j\}\; ge\; n\_\{i+1,j\}\; ,\; .$Thinking of the stack of $n\_\{i,j\}$ unit cubes placed on (i,j)-square, we obtain a "solid" (or 3-dimensional) partition.

Define the sum of the plane partition by:$n=sum\_\{i,j\}\; n\_\{i,j\}\; ,$and let PL("n") denote the number of plane partitions with sum "n".

For example, there are six plane partitions with sum 3::so PL(3) = 6.

Generating function

By a result of Percy MacMahon the generating function for PL("n"), the number of plane partitions of "n", can be calculated by:$sum\_\{n=0\}^\{infty\}\; mbox\{PL\}(n)\; ,\; x^n\; =\; prod\_\{k=1\}^\{infty\}\; frac\{1\}\{(1-x^k)^\{k\; =\; 1+x+3x^2+6x^3+13x^4+24x^5+cdots.$

This results is 2-dimensional analogue of Euler's product formula for the number of integer partitions of "n". There is no analogous formula for partitions in higher dimensions.

MacMahon formula

Denote by $M(a,b,c)$ the number of solid partitions which fit into $a\; imes\; b\; imes\; c$ box. In the planar case, we obtain the binomial coefficients:: MacMahon formula is the multiplicative formula for general values of $M(a,b,c)$:: $M(a,b,c)\; =\; prod\_\{i=1\}^a\; prod\_\{j=1\}^b\; prod\_\{k=1\}^c\; frac\{i+j+k-1\}\{i+j+k-2\}$This formula was obtained by Percy MacMahon and was later rewritten in this form by Ian Macdonald.

References

* P.A. MacMahon, " [http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABU9009 Combinatory analysis] ", 2 vols, Cambridge University Press, 1915-16.
* G. Andrews, "The Theory of Partitions", Cambridge University Press, Cambridge, 1998, ISBN 052163766X
* I.G. Macdonald, "Symmetric Functions and Hall Polynomials", Oxford University Press, Oxford, 1999, ISBN 0198504500

External links

*MathWorld|title=Plane partition|urlname=PlanePartition
*.