Plane partition

A plane partition (parts as heights)

In mathematics, a plane partition (see also solid partition for three-dimensional array) 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.

Generating function

By a result of Percy MacMahon, the generating function for PL(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 is usually referred to as the MacMahon function. This formula is the 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.^{[citation needed]}

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.

References

• G. Andrews, The Theory of Partitions, Cambridge University Press, Cambridge, 1998, ISBN 052163766X
• Bender, Edward A.; Knuth, Donald E. (1972), "Enumeration of plane partitions", Journal of Combinatorial Theory. Series A 13: 40–54, doi:10.1016/0097-3165(72)90007-6, ISSN 1096-0899, MR0299574 
• I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, 1999, ISBN 0198504500
• P.A. MacMahon, Combinatory analysis, 2 vols, Cambridge University Press, 1915-16.

External links

• Weisstein, Eric W., "Plane partition" from MathWorld.
• (sequence A000219 in OEIS).