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Combinatorial number system

Combinatorial number system 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 Combinatorial number system rather than just read about it. In short: In mathematics, and in particular in combinatorics, the combinatorial number system of degree k (for some positive integer k), also referred to as combinadics, or the Macaulay representation of an integer, is a correspondence between natural numbers (taken to include 0) N and k-combinations. The combinations are represented as strictly decreasing sequences ck > ... > c2 > c1 ≥ 0 where each ci corresponds to the inde…

Combinatorial number system — main illustration
Combinatorial number system — illustration

Key takeaways

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

Reference excerpt

In mathematics, and in particular in combinatorics, the combinatorial number system of degree k (for some positive integer k), also referred to as combinadics, or the Macaulay representation of an integer, is a correspondence between natural numbers (taken to include 0) N and k-combinations. The combinations are represented as strictly decreasing sequences ck > ... > c2 > c1 ≥ 0 where each ci corresponds to the index of a chosen element in a given k-combination. Distinct numbers correspond to distinct k-combinations, and produce them in lexicographic order. The numbers less than ( n k ) {\displaystyle {\tbinom {n}{k}}} correspond to all k-combinations of {0, 1, ..., n − 1}. The correspondence does not depend on the size n of the set that the k-combinations are taken from, so it can be interpreted as a map from N to the k-combinations taken from N; in this view the correspondence is a bijection. The number N corresponding to (ck, ..., c2, c1) is given by

N = ( c k k ) + ⋯ + ( c 2 2 ) + ( c 1 1 ) {\displaystyle N={\binom {c_{k}}{k}}+\cdots +{\binom {c_{2}}{2}}+{\binom {c_{1}}{1}}} . The fact that a combination corresponds to a non-negative integer was observed by Lehmer (1964). Indeed, a greedy algorithm finds the k-combination corresponding to N: take ck maximal with ( c k k ) ≤ N {\displaystyle {\tbinom {c_{k}}{k}}\leq N} , then take ck−1 maximal with ( c k − 1 k − 1 ) ≤ N − ( c k k ) {\displaystyle {\tbinom {c_{k-1}}{k-1}}\leq N-{\tbinom {c_{k}}{k}}} , and so forth. Finding the number N, using the formula above, from the k-combination (ck, ..., c2, c1) is also known as "ranking", and the opposite operation (given by the greedy algorithm) as "unranking"; the operations are known by these names in most computer algebra systems, and in computational mathematics. The term "combinatorial representation of integers" was shortened to "combinatorial number system" by Knuth (2011). He also references Ernesto Pascal (1887). The term "combinadic" is introduced by James McCaffrey. Unlike the factorial number system, the combinatorial number system of degree k is not a mixed radix system: the part ( c i i ) {\displaystyle {\tbinom {c_{i}}{i}}} of the number N represented by a "digit" ci is not obtained from it by simply multiplying by a place value. The main application of the combinatorial number system is that it allows rapid computation of the k-combination that is at a given position in the lexicographic ordering, without having to explicitly list the k-combinations preceding it; this allows for instance random generation of k-combinations of a given set. Enumeration of k-combinations has many applications, among which are software testing, sampling, quality control, and the analysis of lottery games.

… excerpt ends here. Continue reading the full article.

Illustrations

Combinatorial number system: The ( 5 3 ) = 10 {\displaystyle {\tbinom {5}{3}}=10} rows of this matrix are the 3-combinations of { 0...4 } {\displaystyle \{0...4\}} . When n {\displaystyle {\color {Red}n}} is the k {\displaystyle {\color {Green}k}} -th element of the combination, the "digit" is ( n k ) {\displaystyle {\tbinom {\color {Red}n}{\color {Green}k}}} .
The ( 5 3 ) = 10 {\displaystyle {\tbinom {5}{3}}=10} rows of this matrix are the 3-combinations of { 0...4 } {\displaystyle \{0...4\}} . When n {\displaystyle {\color {Red}n}} is the k {\displaystyle {\color {Green}k}} -th element of the combination, the "digit" is ( n k ) {\displaystyle {\tbinom {\color {Red}n}{\color {Green}k}}} .
Combinatorial number system: Ernesto Pascal (1887): An integer N {\displaystyle N} has a unique expression as ∑ k = 1 n ( x k k ) {\displaystyle \sum _{k=1}^{n}{\binom {x_{k}}{k}}} with increasing x k {\displaystyle x_{k}} .
Ernesto Pascal (1887): An integer N {\displaystyle N} has a unique expression as ∑ k = 1 n ( x k k ) {\displaystyle \sum _{k=1}^{n}{\binom {x_{k}}{k}}} with increasing x k {\displaystyle x_{k}} .

Worked examples

Example 1 — a first encounter with Combinatorial number system

Start with the simplest possible case. Write down what Combinatorial number system 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 Combinatorial number system 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 Combinatorial number system 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 Combinatorial number system

In research
Combinatorial number system 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 Combinatorial number system 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
Combinatorial number system is common in secondary-school and first-year university syllabi. It links to neighbouring topics Combinatorics, Factorial and binomial topics, so understanding it makes those chapters shorter.
In everyday life
Look for Combinatorial number system 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 Combinatorial number system in 20 minutes

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

Frequently asked questions

What is Combinatorial number system in simple terms?

In mathematics, and in particular in combinatorics, the combinatorial number system of degree k (for some positive integer k), also referred to as combinadics, or the Macaulay representation of an integer, is a correspondence between natural numbers (taken to include 0) N and k-combinations. The co…

Why does Combinatorial number system 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 Combinatorial number system?

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 Combinatorial number system.

Tags

  • Combinatorics
  • Factorial and binomial topics

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