Compositions

Abstract

The compositions of n into at most k parts are the ordered tuples$\left( {x_0, x_1,...x_{k - 1} }\right)$ where x₀+x₁+...+x_k−1=n and 0 ≤ x_i ≤ n. Order matters: one 4-composition of 7 is (0, 1, 5, 1), different ones are (5, 0, 1, 1) and (0, 5, 1, 1). The compositions of n into at most k parts are also called ‘k-compositions of n’. To obtain the compositions of n into exactly k parts (where k ≤ n) generate the compositions of n−k into k parts and add one to each position.