(1)

is sometimes used to explicitly indicate the variable.

Composition is associative, so that

(2)

If the functions is continuous at and is continuous at , then is also continuous at .

Faá 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,

			(3)
			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)

A positive integer has compositions.

The compositions of into parts is given by Compositions[n, k] in the Mathematica add-on package DiscreteMath`Combinatorica` (which can be loaded with the command <<DiscreteMath`) . This command treats 0 as a significant addend, so for example and are considered distinct compositions of length 2. The number of compositions of a number of length is given by the formula

(11)

which implemented as NumberOfCompositions[n, k] in the Mathematica add-on package DiscreteMath`Combinatorica` (which can be loaded with the command <<DiscreteMath`) . The following table gives

	Sloane	, , ...
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.