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Definition & Example
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A plane partition $\Lambda = [\lambda^{(1)},\dots,\lambda^{(k)}]$ is a sequence of integer partitions $\lambda^{(i)}$ such that $\lambda^{(i)} \supseteq \lambda^{(i+1)}$ for all $i$.
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Equivalently, it is an array $\Lambda = [\lambda^{(i)}_j]_{ij}$ of positive integers such that $\lambda^{(i)}_j \geq \lambda^{(i+1)}_j$ and $\lambda^{(i)}_j \geq \lambda^{(i)}_{j+1}$ for all well-defined indices $i,j$.
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$\Lambda$ is a plane partition of $n$, denoted by $\Lambda \vdash n$, if $n = n_1 + \dots + n_k$ for $\lambda^{(i)} \vdash n_i$ being a partition of $n_i$.
| the 6 Plane partitions of size 3 | |||||
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| [[1],[1],[1]] | [[2],[1]] | [[1,1],[1]] | [[3]] | [[2,1]] | [[1,1,1]] |
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Plane partitions are graphically represented by their Ferrers diagram (or Young diagram) as a collection of boxes.
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The number of plane partitions $P(n)$ is A000219. Its generating function is
$$\displaystyle \sum_{n=0}^\infty P(n) x^n = \prod_{k=1}^\infty \frac{1}{(1-x^k)^k}.$$
Properties
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The number of plane partitions inside an $a\times b\times c$ bounding box is given by MacMahon's box formula:
$$\displaystyle \prod_{i=1}^a \displaystyle \prod_{j=1}^b \displaystyle \prod_{k=1}^c \frac{i+j+k-1}{i+j+k-2}.$$ -
There are ten symmetry classes of plane partitions, each of which is enumerated by a nice product formula. These symmetry classes include symmetric, cyclically symmetric, self-complementary, transpose-complementary, and combinations thereof.
References
- Plane partitions in wikipedia.