A composition of a positive integer is an ordered arrangement of positive 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,

			(1)
			(2)
			(3)
			(4)
			(5)
			(6)
			(7)
			(8)

A positive integer has compositions.

The nesting of two or more functions to form a new function is known as function composition.

For integers , the number of compositions of into positive parts is given by

			(9)
			(10)

For , .

A weak composition allows zero as a part. For integers and , the number of weak compositions of of length is given by

			(11)
			(12)

The following table gives counts of weak compositions.

	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.