Plane partition

A plane partition (parts as heights)

In mathematics and especially in combinatorics, a plane partition is a two-dimensional array of nonnegative integers $n_{i,j}$ (with positive integer indices i and j) that is nonincreasing in both indices, that is, that satisfies

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

for all i and j,

and for which only finitely many of the n_i,j are nonzero. A plane partitions may be represented visually by the placement of a stack of $n_{i,j}$ unit cubes above the point (i,j) in the plane, giving a three-dimensional solid like the one shown at right.

The sum of a plane partition is

n=\sum_{i,j} n_{i,j} \,

and PL(n) denotes 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. (Here the plane partitions are drawn using matrix indexing for the coordinates and the entries equal to 0 are suppressed for readability.)

Ferrers diagrams for plane partitions

Another representation for plane partitions is in the form of Ferrers diagrams. The Ferrers diagram of a plane partition of $n$ is a collection of $n$ points or nodes, $\lambda=(\mathbf{y}_1,\mathbf{y}_2,\ldots,\mathbf{y}_n)$ , with $\mathbf{y}_i\in \mathbb{Z}_{\geq0}^{3}$ satisfying the condition:^[1]

Condition FD: If the node

\mathbf{a}=(a_1,a_2,a_3)\in \lambda

, then so do all the nodes

\mathbf{y}=(y_1,y_2,y_3)

with

0\leq y_i\leq a_i

for all

i=1,2,3

Replacing every node of a plane partition by a unit cube with edges aligned with the axes leads to the stack of cubes representation for the plane partition.

Equivalence of the two representations

Given a Ferrers diagram, one constructs the plane partition (as in the main definition) as follows.

Let

n_{i,j}

be the number of nodes in the Ferrers diagram with coordinates of the form

(i-1,j-1,*)

where

*

denotes an arbitrary value. The collection

n_{i,j}

form a plane partition. One can verify that condition FD implies that the conditions for a plane partition are satisfied.

Given a set of $n_{i,j}$ that form a plane partition, one obtains the corresponding Ferrers diagram as follows.

Start with the Ferrers diagram with no nodes. For every non-zero

n_{i,j}

, add

n_{i,j}

nodes of the form

(i-1,j-1,y_3)

for

0\leq y_3< n_{i,j}-1

to the Ferrers diagram. By construction, it is easy to see that condition FD is satisfied.

For instance, below we show the two representations of a plane partitions of 5.

\left( \begin{smallmatrix} 0\\ 0\\ 0 \end{smallmatrix} \begin{smallmatrix} 0\\ 0\\ 1 \end{smallmatrix} \begin{smallmatrix} 0\\ 1\\ 0 \end{smallmatrix} \begin{smallmatrix}1 \\ 0 \\ 0 \end{smallmatrix} \begin{smallmatrix} 1 \\ 1\\ 0 \end{smallmatrix} \right) \qquad \Longleftrightarrow \qquad \begin{matrix} 2 & 1 \\ 1 & 1 \end{matrix}

Above, every node of the Ferrers diagram is written as a column and we have only written only the non-vanishing $n_{i,j}$ as is conventional.

Action of S₃ on plane partitions

There is a natural action of the permutation group $S_3$ on a Ferrers diagram—this corresponds to simultaneously permuting the three coordinates of all nodes. This generalizes the conjugation operation for partitions. The action of $S_3$ can generate new plane partitions starting from a given plane partition. Below we show six plane partitions of 4 that are generated by the $S_3$ action. Only the exchange of the first two coordinates is manifest in the representation given below.

\begin{smallmatrix} 3 & 1 \end{smallmatrix} \quad \begin{smallmatrix} 3 \\ 1 \end{smallmatrix} \quad \begin{smallmatrix} 2 & 1 & 1\end{smallmatrix} \quad \begin{smallmatrix} 2 \\ 1 \\ 1 \end{smallmatrix} \quad \begin{smallmatrix} 1 & 1 & 1 \\ 1 \end{smallmatrix} \quad \begin{smallmatrix} 1 & 1 \\ 1 \\ 1 \end{smallmatrix}

Generating function

By a result of Percy MacMahon, the generating function for PL(n) is given 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.

^[2]

This is sometimes referred to as the MacMahon function.

This formula may be viewed as the 2-dimensional analogue of Euler's product formula for the number of integer partitions of n. There is no analogous formula known for partitions in higher dimensions (i.e., for solid partitions).^[3]

MacMahon formula

Denote by $M(a,b,c)$ the number of plane partitions that fit into $a \times b \times c$ box; that is, the number of plane partitions for which n_i,j ≤ c and n_i,j = 0 whenever i > a or j > b. In the planar case (when c = 1), 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.

Asymptotics of plane partitions

The asymptotics of plane partitions was worked out by E. M. Wright.^[4] One has, for large $n$ :

\mathrm{PL}(n)\sim \frac{ \zeta(3)^{7/36}}{\sqrt{12\pi}}\ \left(\frac{n}{2}\right)^{-25/36} \ \exp\left(3\ \zeta(3)^{1/3} \left(\frac{n}2\right)^{2/3}+ \zeta'(-1)\right)\ ,

where we have corrected for the typographical error (in Wright's paper) pointed out by Mutafchiev and Kamenov.^[5] Evaluating numerically, one finds

n^{-2/3} \ln \mathrm{PL}(n) \sim 2.00945 -0.69444\ n^{-2/3}\ \ln n -1.14631\ n^{-2/3}\ .

Symmetries

Plane partitions may be classified according to various symmetries.^[6] When viewed as a two-dimensional array of integers, there is the natural symmetry of conjugation or transpose that corresponds to switching the indices i and j; for example, the two plane partitions

\begin{matrix} 4 & 2 & 1 \\ 3 & 1\end{matrix}

and

\begin{matrix} 4 & 3 \\ 2 & 1 \\ 1\end{matrix}

are conjugate. When viewed as three-dimensional arrays of blocks, however, more symmetries become evident: any permutation of the axes corresponds to a reflection or rotation of the plane partition. A plane partition that is invariant under all of these symmetries is called totally symmetric.

An additional symmetry is complementation: given a plane partition inside an $a \times b \times c$ box, the complement is simply the result of removing the boxes of the plane partition from the box and reindexing appropriately. Totally symmetric plane partitions that are equal to their own complements are known as totally symmetric self-complementary plane partitions; they are known to be equinumerous with alternating sign matrices and so with numerous other combinatorial objects.

References

↑ A. O. L. Atkin, P. Bratley, I. G. Macdonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097–1100.
↑ R.P. Stanley, Enumerative Combinatorics, Volume 2. Corollary 7.20.3.
↑ R.P. Stanley, Enumerative Combinatorics, Volume 2. pp. 365, 401–2.
↑ E. M. Wright, Asymptotic partition formulae I. Plane partitions, The Quarterly Journal of Mathematics 1 (1931) 177–189.
↑ L. Mutafchiev and E. Kamenov, "Asymptotic formula for the number of plane partitions of positive integers", Comptus Rendus-Academie Bulgare Des Sciences 59 (2006), no. 4, 361.
↑ R.P. Stanley, "Symmetries of plane partitions", J. Combinatorial Theory (A) 43 (1986), 103-113. Erratum, 44 (1987), 310.

G. Andrews, The Theory of Partitions, Cambridge University Press, Cambridge, 1998, ISBN 0-521-63766-X
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, MR 0299574
I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, 1999, ISBN 0-19-850450-0
P.A. MacMahon, Combinatory analysis, 2 vols, Cambridge University Press, 1915-16.

External links

Weisstein, Eric W., "Plane partition", MathWorld.
(sequence A000219 in OEIS).
The DLMF page on Plane Partitions