compositionsuch

Compositionsuch is a term used in theoretical discussions to denote a constrained form of an integer composition. In this sense, a composition of a positive integer n is called compositionsuch with respect to a rule set R if every adjacent pair of parts satisfies a relation encoded by R. The concept is used as a generic model for studying how local connection rules affect global structure.

Let n be a positive integer and c = (c1, c2, ..., ck) a composition of n, meaning sum ci = n and each ci ≥ 1. The composition c is compositionsuch with respect to R ⊆ N×N, a relation on positive integers, if for every i = 1,...,k-1, (ci, ci+1) ∈ R. If R is the universal relation, every composition is compositionsuch; for other R, only a subset qualifies.

Example: take n = 5 and R = { (a,b) : a ≥ b }. The compositions of 5 that are compositionsuch with respect to R include (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), and (1,1,1,1,1); the sequence (1,4) is not compositionsuch since 1 is not ≥ 4.

Enumeration and detection of compositionsuch instances can be approached by dynamic programming or automata methods, especially when R is finite or has a regular structure. Generating functions can encode the allowed transitions to count all compositions of n that satisfy the rule.

The term is not standard in published literature and is typically used as a didactic placeholder to discuss constrained compositions. See also: integer composition, constrained composition, generating function.