Composition (number Theory)
In mathematics, a composition of an integer n is a way of writing n as the sum of a sequence of (strictly) 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 partition of that number. Any integer has finitely many distinct compositions. Negative numbers do not have any compositions, but 0 has one composition, the empty sequence. Any positive integer n has 2n−1 distinct compositions. This is a power of two, because every composition matches a binary number.
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 any 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.
Read more about Composition (number Theory): Examples, Number of Compositions
Famous quotes containing the word composition:
“There is singularly nothing that makes a difference a difference in beginning and in the middle and in ending except that each generation has something different at which they are all looking. By this I mean so simply that anybody knows it that composition is the difference which makes each and all of them then different from other generations and this is what makes everything different otherwise they are all alike and everybody knows it because everybody says it.”
—Gertrude Stein (1874–1946)