The nesting of two or more functions to form a single new function is known as composition. The composition of two functions and is denoted , where is a function whose domain includes the range of . The notation
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is sometimes used to explicitly indicate the variable.
Composition is associative, so that
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If the functions is continuous at and is continuous at , then is also continuous at .
A function which is the composition of two other functions, say and , is sometimes said to be a composite function.
Faà di Bruno's formula gives an explicit formula for the th derivative of the composition .
A combinatorial composition is defined as an ordered arrangement of nonnegative integers which sum to (Skiena 1990, p. 60). It is therefore a partition in which order is significant. For example, there are eight compositions of 4,
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A positive integer has compositions.
The number of compositions of into parts (where 0 is not allowed as a part) is given by
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The number of compositions of a number of length (where 0 is allowed) is given by the formula
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which is implemented as NumberOfCompositions[n, k] in the Wolfram Language package Combinatorica` . The following table gives counts of compositions where 0 is allowable.
OEIS | , , ... | |
2 | A000027 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, ... |
3 | A000217 | 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, ... |
4 | A000292 | 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, ... |
5 | A000332 | 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365, 1820, ... |
6 | A000389 | 6, 21, 56, 126, 252, 462, 792, 1287, 2002, 3003, 4368, ... |
7 | A000579 | 7, 28, 84, 210, 462, 924, 1716, 3003, 5005, 8008, 12376, ... |
8 | A000580 | 8, 36, 120, 330, 792, 1716, 3432, 6435, 11440, 19448, ... |
9 | A000581 | 9, 45, 165, 495, 1287, 3003, 6435, 12870, 24310, 43758, ... |
An operation called composition is also defined on binary quadratic forms. For two numbers represented by two forms, the product can then be represented by the composition. For example, the composition of the forms and is given by , and in this case, the product of 17 and 13 would be represented as (). There are several algorithms for computing binary quadratic form composition, which is the basis for some factoring methods.