mathematics

Lah number

Lah number is a mathematics 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 Lah number rather than just read about it. In short: In mathematics, the (signed and unsigned) Lah numbers are coefficients expressing rising factorials in terms of falling factorials and vice versa. They were discovered by Ivo Lah in 1954.

Key takeaways

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

Reference excerpt

In mathematics, the (signed and unsigned) Lah numbers are coefficients expressing rising factorials in terms of falling factorials and vice versa. They were discovered by Ivo Lah in 1954. Explicitly, the unsigned Lah numbers L ( n , k ) {\displaystyle L(n,k)} are given by the formula involving the binomial coefficient

L ( n , k ) = ( n − 1 k − 1 ) n ! k ! {\displaystyle L(n,k)={n-1 \choose k-1}{\frac {n!}{k!}}}

for n ≥ k ≥ 1 {\displaystyle n\geq k\geq 1} , and the signed Lah numbers L ′ ( n , k ) {\displaystyle L'(n,k)} are related to them by L ′ ( n , k ) = ( − 1 ) n L ( n , k ) {\displaystyle L'(n,k)=(-1)^{n}L(n,k)} . Signed Lah numbers are only of historical interest as it's how they were defined in Lah's seminal paper, but their sign pattern ( ( − 1 ) n {\displaystyle (-1)^{n}} , instead of ( − 1 ) n − k {\displaystyle (-1)^{n-k}} as used for signed Stirling numbers) make them of little to no use in formulas of mathematical interest. Unsigned Lah numbers have an interesting meaning in combinatorics: they count the number of ways a set of n {\textstyle n} elements can be partitioned into k {\textstyle k} nonempty linearly ordered subsets. Lah numbers are related to Stirling numbers. For n ≥ 1 {\textstyle n\geq 1} , the Lah number L ( n , 1 ) {\textstyle L(n,1)} is equal to the factorial n ! {\textstyle n!} in the interpretation above, the only partition of { 1 , 2 , 3 } {\textstyle \{1,2,3\}} into 1 set can have its set ordered in 6 ways: { ( 1 , 2 , 3 ) } , { ( 1 , 3 , 2 ) } , { ( 2 , 1 , 3 ) } , { ( 2 , 3 , 1 ) } , { ( 3 , 1 , 2 ) } , { ( 3 , 2 , 1 ) } {\displaystyle \{(1,2,3)\},\{(1,3,2)\},\{(2,1,3)\},\{(2,3,1)\},\{(3,1,2)\},\{(3,2,1)\}}

L ( 3 , 2 ) {\textstyle L(3,2)} is equal to 6, because there are six partitions of { 1 , 2 , 3 } {\textstyle \{1,2,3\}} into two ordered parts: { 1 , ( 2 , 3 ) } , { 1 , ( 3 , 2 ) } , { 2 , ( 1 , 3 ) } , { 2 , ( 3 , 1 ) } , { 3 , ( 1 , 2 ) } , { 3 , ( 2 , 1 ) } {\displaystyle \{1,(2,3)\},\{1,(3,2)\},\{2,(1,3)\},\{2,(3,1)\},\{3,(1,2)\},\{3,(2,1)\}}

L ( n , n ) {\textstyle L(n,n)} is always 1 because the only way to partition { 1 , 2 , … , n } {\textstyle \{1,2,\ldots ,n\}} into n {\displaystyle n} non-empty subsets results in subsets of size 1, that can only be permuted in one way. In the more recent literature, Karamata–Knuth style notation has taken over. Lah numbers are now often written as L ( n , k ) = ⌊ n k ⌋ {\displaystyle L(n,k)=\left\lfloor {n \atop k}\right\rfloor }

Table of values Below is a table of values for the Lah numbers:

The row sums are 1 , 1 , 3 , 13 , 73 , 501 , 4051 , 37633 , … {\textstyle 1,1,3,13,73,501,4051,37633,\dots } (sequence A000262 in the OEIS).

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Lah number

Start with the simplest possible case. Write down what Lah number claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Lah number 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 Lah number 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 Lah number

In research: Lah number appears in mathematics 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 Lah number 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: Lah number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Factorial and binomial topics, Integer sequences, Triangles of numbers, so understanding it makes those chapters shorter.
In everyday life: Look for Lah number 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 Lah number in 20 minutes

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

Frequently asked questions

What is Lah number in simple terms?

In mathematics, the (signed and unsigned) Lah numbers are coefficients expressing rising factorials in terms of falling factorials and vice versa. They were discovered by Ivo Lah in 1954.

Why does Lah number matter?

Because it connects several mathematics 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 Lah number?

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 Lah number.