Plane partition

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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

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.

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.

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

Eric W. Weisstein, Plane partition at MathWorld.
(sequence A000219 in OEIS).

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