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Plane partition

Plane partition is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Plane partition rather than just read about it. In short: In mathematics and especially in combinatorics, a plane partition is a two-dimensional array of nonnegative integers π i , j {\displaystyle \pi _{i,j}} (with positive integer indices i and j) that is nonincreasing in both indices. This means that π i , j ≥ π i , j + 1 {\displaystyle \pi _{i,j}\geq \pi _{i,j+1}} and π i , j ≥ π i + 1 , j {\displaystyle \pi _{i,j}\geq \pi _{i+1,j}} for all i and j.

Plane partition — main illustration
Plane partition — illustration

Key takeaways

  • Plane partition belongs to science; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Plane partition to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Plane partition from memory before moving on to harder problems.

Reference excerpt

In mathematics and especially in combinatorics, a plane partition is a two-dimensional array of nonnegative integers π i , j {\displaystyle \pi _{i,j}} (with positive integer indices i and j) that is nonincreasing in both indices. This means that

π i , j ≥ π i , j + 1 {\displaystyle \pi _{i,j}\geq \pi _{i,j+1}} and π i , j ≥ π i + 1 , j {\displaystyle \pi _{i,j}\geq \pi _{i+1,j}} for all i and j. Moreover, only finitely many of the π i , j {\displaystyle \pi _{i,j}} may be nonzero. Plane partitions are a generalization of partitions of an integer. A plane partition may be represented visually by the placement of a stack of π i , j {\displaystyle \pi _{i,j}} unit cubes above the point (i, j) in the plane, giving a three-dimensional solid as shown in the picture. The image has matrix form

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

Plane partitions are also often described by the positions of the unit cubes. From this point of view, a plane partition can be defined as a finite subset P {\displaystyle {\mathcal {P}}} of positive integer lattice points (i, j, k) in N 3 {\displaystyle \mathbb {N} ^{3}} , such that if (r, s, t) lies in P {\displaystyle {\mathcal {P}}} and if ( i , j , k ) {\displaystyle (i,j,k)} satisfies 1 ≤ i ≤ r {\displaystyle 1\leq i\leq r} , 1 ≤ j ≤ s {\displaystyle 1\leq j\leq s} , and 1 ≤ k ≤ t {\displaystyle 1\leq k\leq t} , then (i, j, k) also lies in P {\displaystyle {\mathcal {P}}} . The sum of a plane partition is

n = ∑ i , j π i , j . {\displaystyle n=\sum _{i,j}\pi _{i,j}.}

The sum describes the number of cubes of which the plane partition consists. Much interest in plane partitions concerns the enumeration of plane partitions in various classes. The number of plane partitions with sum n is denoted by PL(n). For example, there are six plane partitions with sum 3

3 2 1 1 1 1 2 1 1 1 1 1 1 1 {\displaystyle {\begin{matrix}3\end{matrix}}\qquad {\begin{matrix}2&1\end{matrix}}\qquad {\begin{matrix}1&1&1\end{matrix}}\qquad {\begin{matrix}2\\1\end{matrix}}\qquad {\begin{matrix}1&1\\1\end{matrix}}\qquad {\begin{matrix}1\\1\\1\end{matrix}}}

so PL(3) = 6. Plane partitions may be classified by how symmetric they are. Many symmetric classes of plane partitions are enumerated by simple product formulas.

Generating function of plane partitions The generating function for PL(n) is

… excerpt ends here. Continue reading the full article.

Illustrations

Plane partition: A plane partition of 30 represented as stacks of unit cubes
A plane partition of 30 represented as stacks of unit cubes
Plane partition: A symmetric plane partition, sum 35
A symmetric plane partition, sum 35
Plane partition: A cyclically symmetric plane partition
A cyclically symmetric plane partition
Plane partition: A totally symmetric plane partition
A totally symmetric plane partition
Plane partition: A self-complementary plane partition
A self-complementary plane partition

Worked examples

Example 1 — a first encounter with Plane partition

Start with the simplest possible case. Write down what Plane partition claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Plane partition before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Plane partition ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Plane partition

In research
Plane partition appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Plane partition in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Plane partition is common in secondary-school and first-year university syllabi. It links to neighbouring topics Enumerative combinatorics, Integer partitions, so understanding it makes those chapters shorter.
In everyday life
Look for Plane partition outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Plane partition in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Plane partition means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Plane partition out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Plane partition in simple terms?

In mathematics and especially in combinatorics, a plane partition is a two-dimensional array of nonnegative integers π i , j {\displaystyle \pi _{i,j}} (with positive integer indices i and j) that is nonincreasing in both indices. This means that π i , j ≥ π i , j + 1 {\displaystyle \pi _{i,j}\geq…

Why does Plane partition matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Plane partition?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Plane partition.

Tags

  • Enumerative combinatorics
  • Integer partitions

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