enumeration

Enumeration is the act of listing elements of a set or counting them, typically with the goal of completeness. In mathematics, enumeration often refers to enumerative combinatorics, the study of counting discrete structures and listing all objects of a given type. The central aim is to determine, for each size n, how many objects satisfy certain conditions, and sometimes to describe all objects explicitly. Techniques include basic counting principles (sum and product rules), permutations and combinations, partitions, and more advanced methods such as generating functions, recurrences, and polynomials. Generating functions encode counting sequences as coefficients of power series and are used to derive formulas and asymptotics.

In computer science, enumeration also denotes algorithms that generate every object of a class, for example all graphs on n labeled vertices or all feasible schedules. Such exhaustive enumeration supports testing, verification, and optimization, though it may be computationally expensive. The term also appears as a data type: an enumeration (enum) is a finite set of named values used to represent a fixed set of constants in programming languages.

In logic and computability theory, a set is recursively enumerable if there exists an algorithm that lists its members, possibly with repetition and without a known termination for non-members. This notion contrasts with decidable sets, where membership can be determined in finite time. Recursively enumerable sets play a central role in topics such as undecidability and complexity theory.