Plane partition

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:: egin{matrix} 1 & 1 & 1 end{matrix} qquad egin{matrix} 1 & 1 \ 1 & end{matrix}qquad egin{matrix} 1 \ 1 \ 1 & end{matrix}qquad egin{matrix} 2 & 1 & end{matrix}qquad egin{matrix} 2 \ 1 & end{matrix}qquad egin{matrix} 3 end{matrix}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:: M(a,b,1) = inom{a+b}{a}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
*.


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