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Lattice path

Lattice path 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 Lattice path rather than just read about it. In short: In combinatorics, a lattice path L in the d-dimensional integer lattice ⁠ Z d {\displaystyle \mathbb {Z} ^{d}} ⁠ of length k with steps in the set S, is a sequence of vectors ⁠ v 0 , v 1 , … , v k ∈ Z d {\displaystyle v_{0},v_{1},\ldots ,v_{k}\in \mathbb {Z} ^{d}} ⁠ such that each consecutive difference v i − v i − 1 {\displaystyle v_{i}-v_{i-1}} lies in S. A lattice path may lie in any lattice in ⁠ R d {\displaysty…

Lattice path — main illustration
Lattice path — illustration

Key takeaways

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

Reference excerpt

In combinatorics, a lattice path L in the d-dimensional integer lattice ⁠ Z d {\displaystyle \mathbb {Z} ^{d}} ⁠ of length k with steps in the set S, is a sequence of vectors ⁠ v 0 , v 1 , … , v k ∈ Z d {\displaystyle v_{0},v_{1},\ldots ,v_{k}\in \mathbb {Z} ^{d}} ⁠ such that each consecutive difference v i − v i − 1 {\displaystyle v_{i}-v_{i-1}} lies in S. A lattice path may lie in any lattice in ⁠ R d {\displaystyle \mathbb {R} ^{d}} ⁠, but the integer lattice ⁠ Z d {\displaystyle \mathbb {Z} ^{d}} ⁠ is most commonly used. An example of a lattice path in ⁠ Z 2 {\displaystyle \mathbb {Z} ^{2}} ⁠ of length 5 with steps in S = { ( 2 , 0 ) , ( 1 , 1 ) , ( 0 , − 1 ) } {\displaystyle S=\lbrace (2,0),(1,1),(0,-1)\rbrace }

is L = { ( − 1 , − 2 ) , ( 0 , − 1 ) , ( 2 , − 1 ) , ( 2 , − 2 ) , ( 2 , − 3 ) , ( 4 , − 3 ) } {\displaystyle L=\lbrace (-1,-2),(0,-1),(2,-1),(2,-2),(2,-3),(4,-3)\rbrace } .

North-East lattice paths A North-East (NE) lattice path is a lattice path in Z 2 {\displaystyle \mathbb {Z} ^{2}} with steps in S = { ( 0 , 1 ) , ( 1 , 0 ) } {\displaystyle S=\lbrace (0,1),(1,0)\rbrace } . The ( 0 , 1 ) {\displaystyle (0,1)} steps are called North steps and denoted by N {\displaystyle N} s; the ( 1 , 0 ) {\displaystyle (1,0)} steps are called East steps and denoted by E {\displaystyle E} s. NE lattice paths most commonly begin at the origin. This convention allows encoding all the information about a NE lattice path L {\displaystyle L} in a single permutation word. The length of the word gives the number of steps of the lattice path, k {\displaystyle k} . The order of the N {\displaystyle N} s and E {\displaystyle E} s communicates the sequence of L {\displaystyle L} . Furthermore, the number of N {\displaystyle N} s and the number of E {\displaystyle E} s in the word determines the end point of L {\displaystyle L} . If the permutation word for a NE lattice path contains n {\displaystyle n} N {\displaystyle N} -steps and e {\displaystyle e} E {\displaystyle E} -steps, and if the path begins at the origin, then the path necessarily ends at ( e , n ) {\displaystyle (e,n)} . This follows because the path "walks" exactly n {\displaystyle n} steps North and e {\displaystyle e} steps East from ( 0 , 0 ) {\displaystyle (0,0)} .

Counting lattice paths Lattice paths are often used to count other combinatorial objects. Similarly, there are many combinatorial objects that count the number of lattice paths of a certain kind. This occurs when the lattice paths are in bijection with the object in question. For example,

… excerpt ends here. Continue reading the full article.

Illustrations

Lattice path: Lattice path of length 5 in ℤ2 with S = { (2,0), (1,1), (0,-1) }.
Lattice path of length 5 in ℤ2 with S = { (2,0), (1,1), (0,-1) }.
Lattice path: The four NE lattice paths starting from ( 0 , 0 ) {\displaystyle (0,0)} with exactly one N {\displaystyle N} and three E {\displaystyle E} s. The endpoint is necessarily at ( 3 , 1 ) {\displaystyle (3,1)} .
The four NE lattice paths starting from ( 0 , 0 ) {\displaystyle (0,0)} with exactly one N {\displaystyle N} and three E {\displaystyle E} s. The endpoint is necessarily at ( 3 , 1 ) {\displaystyle (3,1)} .
Lattice path: The number of lattice paths from ( 0 , 0 ) {\displaystyle (0,0)} to ( 2 , 3 ) {\displaystyle (2,3)} is equal to ( 2 + 3 2 ) = ( 5 2 ) = 10 {\displaystyle {\binom {2+3}{2}}={\binom {5}{2}}=10} .
The number of lattice paths from ( 0 , 0 ) {\displaystyle (0,0)} to ( 2 , 3 ) {\displaystyle (2,3)} is equal to ( 2 + 3 2 ) = ( 5 2 ) = 10 {\displaystyle {\binom {2+3}{2}}={\binom {5}{2}}=10} .
Lattice path: Each NE lattice path passes through exactly one colored node.
Each NE lattice path passes through exactly one colored node.
Lattice path: Sets of NE lattice paths squared, with the second copy rotated 90° clockwise.
Sets of NE lattice paths squared, with the second copy rotated 90° clockwise.

Worked examples

Example 1 — a first encounter with Lattice path

Start with the simplest possible case. Write down what Lattice path 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 Lattice path 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 Lattice path 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 Lattice path

In research
Lattice path 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 Lattice path 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
Lattice path is common in secondary-school and first-year university syllabi. It links to neighbouring topics Enumerative combinatorics, so understanding it makes those chapters shorter.
In everyday life
Look for Lattice path 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 Lattice path in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Lattice path 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 Lattice path out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Lattice path in simple terms?

In combinatorics, a lattice path L in the d-dimensional integer lattice ⁠ Z d {\displaystyle \mathbb {Z} ^{d}} ⁠ of length k with steps in the set S, is a sequence of vectors ⁠ v 0 , v 1 , … , v k ∈ Z d {\displaystyle v_{0},v_{1},\ldots ,v_{k}\in \mathbb {Z} ^{d}} ⁠ such that each consecutive diffe…

Why does Lattice path 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 Lattice path?

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 Lattice path.

Tags

  • Enumerative combinatorics

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