compositions

Compositions in mathematics refers to an ordered way of writing a positive integer as a sum of positive integers. A composition of n is a sequence (a1, a2, ..., ak) with ai > 0 and a1 + a2 + ... + ak = n. The order of parts matters, unlike in partitions. For example, 4 has compositions such as 4, 3+1, 1+3, 2+2, 2+1+1, 1+2+1, 1+1+2, and 1+1+1+1. The number of compositions of n is 2^(n-1).

In music, a composition is a musical work created by a composer. It may be instrumental or vocal and is usually notated with performance instructions. Compositions span from short pieces to large cycles and cover genres such as classical, jazz, and contemporary music.

In visual arts and design, composition refers to the arrangement of visual elements within a work. Principles include balance, contrast, emphasis, rhythm, proportion, and unity. Guidelines such as the rule of thirds or the golden ratio are often discussed, though artists may deviate from them.

In writing, a composition is a structured piece of prose, such as an essay, story, or report. In education, students write compositions to demonstrate language skill and organization.

In science, chemical composition describes the makeup of a substance—the elements or compounds present and their proportions. It can be expressed as a formula, a percent composition, or descriptive terms, and it influences properties, reactivity, and classification.