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| [[File:Partition3D.svg|thumb|right|A plane partition (parts as heights)]]
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| In [[mathematics]] and especially in [[combinatorics]], a '''plane partition''' is a two-dimensional array of nonnegative integers <math>n_{i,j}</math> (with [[positive]] [[integer]] indices ''i'' and ''j'') that is nonincreasing in both indices, that is, that satisfies
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| :<math> n_{i,j} \ge n_{i,j+1} \quad\mbox{and}\quad n_{i,j} \ge n_{i+1,j} \,</math> for all ''i'' and ''j'',
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| and for which only finitely many of the ''n''<sub>''i'',''j''</sub> are nonzero. A plane partitions may be represented visually by the placement of a stack of <math>n_{i,j}</math> unit cubes above the point (''i'',''j'') in the plane, giving a three-dimensional solid like the one shown at right.
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| The ''sum'' of a plane partition is | |
| :<math> n=\sum_{i,j} n_{i,j} \, </math>
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| and PL(''n'') denotes the number of plane partitions with sum ''n''. | |
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| For example, there are six plane partitions with sum 3:
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| :<math> \begin{matrix} 1 & 1 & 1 \end{matrix}
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| \qquad \begin{matrix} 1 & 1 \\ 1 & \end{matrix}
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| \qquad \begin{matrix} 1 \\ 1 \\ 1 & \end{matrix}
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| \qquad \begin{matrix} 2 & 1 & \end{matrix}
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| \qquad \begin{matrix} 2 \\ 1 & \end{matrix}
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| \qquad \begin{matrix} 3 \end{matrix}
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| </math> | |
| so PL(3) = 6. (Here the plane partitions are drawn using [[matrix index]]ing for the coordinates and the entries equal to 0 are suppressed for readability.)
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| == Ferrers diagrams for plane partitions ==
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| Another representation for plane partitions is in the form of [[Norman_Macleod_Ferrers|Ferrers]] diagrams. The Ferrers diagram of a plane partition of <math>n</math> is a collection of <math>n</math> points or ''nodes'', <math>\lambda=(\mathbf{y}_1,\mathbf{y}_2,\ldots,\mathbf{y}_n)</math>, with <math>\mathbf{y}_i\in \mathbb{Z}_{\geq0}^{3}</math> satisfying the condition:<ref name="Atkin1967">A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for ''m''-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097–1100.</ref>
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| :'''Condition FD:''' If the node <math>\mathbf{a}=(a_1,a_2,a_3)\in \lambda</math>, then so do all the nodes <math>\mathbf{y}=(y_1,y_2,y_3)</math> with <math>0\leq y_i\leq a_i</math> for all <math>i=1,2,3</math>.
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| 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.
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| === Equivalence of the two representations ===
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| Given a Ferrers diagram, one constructs the solid partition (as in the main definition) as follows.
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| :Let <math>n_{i,j}</math> be the number of nodes in the Ferrers diagram with coordinates of the form <math>(i-1,j-1,,*)</math> where <math>*</math> denotes an arbitrary value. The collection <math>n_{i,j}</math> form a plane partition. One can verify that condition FD implies that the conditions for a plane partition are satisfied.
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| Given a set of <math>n_{i,j}</math> that form a plane partition, one obtains the corresponding Ferrers diagram as follows.
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| :Start with the Ferrers diagram with no nodes. For every non-zero <math>n_{i,j,k}</math>, add <math>n_{i,j,k}</math> nodes <math>(i-1,j-1,y_3)</math> for <math>0\leq y_3< n_{i,j}</math> to the Ferrers diagram. By construction, it is easy to see that condition FD is satisfied.
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| For instance, below we show the two representations of a plane partitions of 5.
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| :<math>
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| \left( \begin{smallmatrix} 0\\ 0\\ 0 \end{smallmatrix}
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| \begin{smallmatrix} 0\\ 0\\ 1 \end{smallmatrix}
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| \begin{smallmatrix} 0\\ 1\\ 0 \end{smallmatrix}
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| \begin{smallmatrix}1 \\ 0 \\ 0 \end{smallmatrix}
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| \begin{smallmatrix} 1 \\ 1\\ 0 \end{smallmatrix}
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| \right) \qquad \Longleftrightarrow \qquad \begin{matrix} 2 & 1 \\ 1 & 1 \end{matrix}
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| </math>
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| Above, every node of the Ferrers diagram is written as a column and we have only written only the non-vanishing <math>n_{i,j}</math> as is conventional.
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| === Action of ''S''<sub>3</sub> on plane partitions ===
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| There is a natural action of the permutation group <math>S_3</math> 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 <math>S_3</math> can generate new plane partitions starting from a given plane partition. Below we show six plane partitions of 4 that are generated by the <math>S_3</math> action. Only the exchange of the first two coordinates is manifest in the representation given below.
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| :<math>
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| \begin{smallmatrix} 3 & 1 \end{smallmatrix} \quad
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| \begin{smallmatrix} 3 \\ 1 \end{smallmatrix} \quad
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| \begin{smallmatrix} 2 & 1 & 1\end{smallmatrix} \quad
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| \begin{smallmatrix} 2 \\ 1 \\ 1 \end{smallmatrix} \quad
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| \begin{smallmatrix} 1 & 1 & 1 \\ 1 \end{smallmatrix} \quad
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| \begin{smallmatrix} 1 & 1 \\ 1 \\ 1 \end{smallmatrix}
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| </math>
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| ==Generating function==
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| By a result of [[Percy Alexander MacMahon|Percy MacMahon]], the [[generating function]] for PL(''n'') is given by
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| :<math> \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. </math><ref>[[Richard P. Stanley|R.P. Stanley]], ''Enumerative Combinatorics'', Volume 2. Corollary 7.20.3.</ref>
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| This is sometimes referred to as the ''MacMahon function''.
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| This formula may be viewed as the 2-dimensional analogue of [[Leonhard Euler|Euler]]'s [[Integer partition#Generating function|product formula]] for the number of [[Partition (number theory)|integer partitions]] of ''n''. There is no analogous formula known for partitions in higher dimensions (i.e., for [[solid partition]]s).<ref>[[Richard P. Stanley|R.P. Stanley]], ''Enumerative Combinatorics'', Volume 2. pp. 365, 401–2.</ref>
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| ==MacMahon formula==
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| Denote by <math>M(a,b,c)</math> the number of plane partitions that fit into <math>a \times b \times c</math> box; that is, the number of plane partitions for which ''n''<sub>''i'',''j''</sub> ≤ ''c'' and ''n''<sub>''i'',''j''</sub> = 0 whenever ''i'' > ''a'' or ''j'' > ''b''. In the planar case (when ''c'' = 1), we obtain the [[binomial coefficient]]s:
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| : <math>M(a,b,1) = \binom{a+b}{a}.</math> | |
| '''MacMahon formula''' is the multiplicative formula for general values of <math>M(a,b,c)</math>:
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| : <math>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}.</math>
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| This formula was obtained by [[Percy Alexander MacMahon|Percy MacMahon]] and was later rewritten in this form by [[Ian G. Macdonald|Ian Macdonald]].
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| == Asymptotics of plane partitions ==
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| The asymptotics of plane partitions was worked out by [[E. M. Wright]].<ref>[[E. M. Wright]], Asymptotic partition formulae I. Plane partitions, The Quarterly Journal of Mathematics '''1''' (1931) 177–189.</ref> One has, for large <math> n </math>:
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| :<math>
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| \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)\ ,
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| </math>
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| where we have corrected for the typographical error (in Wright's paper) pointed out by Mutafchiev and Kamenov.<ref>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.</ref> Evaluating numerically, one finds
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| :<math>
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| n^{-2/3} \ln \mathrm{PL}(n) \sim 2.00945 -0.69444\ n^{-2/3}\ \ln n -1.14631\ n^{-2/3}\ .
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| </math>
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| ==Symmetries==
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| {{Expand section|date=June 2012}}
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| Plane partitions may be classified according to various symmetries.<ref>
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| [[Richard P. Stanley|R.P. Stanley]], "Symmetries of plane partitions", J. Combinatorial Theory (A) '''43''' (1986), 103-113. Erratum, '''44''' (1987), 310. </ref> 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
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| : <math>\begin{matrix} 4 & 2 & 1 \\ 3 & 1\end{matrix}</math> and <math>\begin{matrix} 4 & 3 \\ 2 & 1 \\ 1\end{matrix}</math>
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| 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''.
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| An additional symmetry is ''complementation'': given a plane partition inside an <math>a \times b \times c</math> 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.
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| ==References==
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| <references/>
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| * [[George Andrews (mathematician)|G. Andrews]], ''The Theory of Partitions'', Cambridge University Press, Cambridge, 1998, ISBN 0-521-63766-X
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| *{{Citation | last1=Bender | first1=Edward A. | last2=Knuth | first2=Donald E. | author2-link=Donald Knuth | title=Enumeration of plane partitions | doi=10.1016/0097-3165(72)90007-6 | id={{MathSciNet | id = 0299574}} | year=1972 | journal=Journal of Combinatorial Theory. Series A | issn=1096-0899 | volume=13 | pages=40–54}}
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| * [[Ian G. Macdonald|I.G. Macdonald]], ''Symmetric Functions and Hall Polynomials'', Oxford University Press, Oxford, 1999, ISBN 0-19-850450-0
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| * [[Percy Alexander MacMahon|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.
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| ==External links==
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| *{{MathWorld|title=Plane partition|urlname=PlanePartition}}
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| *{{OEIS|id=A000219}}.
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| * [http://dlmf.nist.gov/26.12 The DLMF page on Plane Partitions ]
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| [[Category:Enumerative combinatorics]]
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