Plane partition

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A 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:

$n_{i,j} \ge n_{i,j+1} \quad\mbox{and}\quad n_{i,j} \ge n_{i+1,j} \, .$

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:

$\begin{matrix} 1 & 1 & 1 \end{matrix} \qquad \begin{matrix} 1 & 1 \\ 1 & \end{matrix} \qquad \begin{matrix} 1 \\ 1 \\ 1 & \end{matrix} \qquad \begin{matrix} 2 & 1 & \end{matrix} \qquad \begin{matrix} 2 \\ 1 & \end{matrix} \qquad \begin{matrix} 3 \end{matrix}$

so PL(3) = 6.

1 Generating function
2 MacMahon formula
3 References
4 External links

[edit] 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.

[edit] MacMahon formula

Denote by $M (a, b, c)$ the number of solid partitions which fit into $a \times b \times c$ box. In the planar case, we obtain the binomial coefficients:

$M(a,b,1) = \binom{a+b}{a}$

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.

[edit] References

P.A. MacMahon, 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