Lesson plan
Composition (combinatorics)
In mathematics, a composition of an integer n is a way of writing n as the sum of a sequence of positive integers. Two sequences that differ in the order of their terms define different compositions of their sum, while they are considered to define the same integer partition of that number. Every integer has finitely many distinct compositions. Negative numbers do not have any compositions, but 0 has one composition, the empty sequence. Each positive integer n has 2 n−1 distinct compositions. between 3 bit binary numbers and compositions of 4 A weak composition of an integer n is similar to a composition of n, but allowing terms of the sequence to be zero: it is a way of writing n as the sum of a sequence of non-negative integers. As a consequence every positive integer admits infinitely many weak compositions (if their length is not bounded). Adding a number of terms 0 to the end of a weak composition is usually not considered to define a different weak composition; in other words, weak compositions are assumed to be implicitly extended indefinitely by terms 0. To further generalize, an A-restricted composition of an integer n, for a subset A of the (nonnegative or positive) integers, is an ordered collection of one or more elements in A whose sum is n.
Article summary
In mathematics, a composition of an integer n is a way of writing n as the sum of a sequence of positive integers. Two sequences that differ in the order of their terms define different compositions of their sum, while they are considered to define the same integer partition of that number. Every integer has finitely many distinct compositions. Negative numbers do not have any compositions, but 0 has one composition, the empty sequence. Each positive integer n has 2 n−1 distinct compositions. between 3 bit binary numbers and compositions of 4 A weak composition of an integer n is similar to a composition of n, but allowing terms of the sequence to be zero: it is a way of writing n as the sum of a sequence of non-negative integers. As a consequence every positive integer admits infinitely many weak compositions (if their length is not bounded). Adding a number of terms 0 to the end of a weak composition is usually not considered to define a different weak composition; in other words, weak compositions are assumed to be implicitly extended indefinitely by terms 0. To further generalize, an A-restricted composition of an integer n, for a subset A of the (nonnegative or positive) integers, is an ordered collection of one or more elements in A whose sum is n.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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