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However, these definitions characterize distinct classes since there are uncountably many subsets of the natural numbers that can be enumerated by an arbitrary function with domain ω and only countably many computable functions. A specific example of a set with an enumeration but not a computable enumeration is the complement of the halting set.
Enumeration problems have been studied in the context of computational complexity theory, and several complexity classes have been introduced for such problems.. A very general such class is EnumP, [1] the class of problems for which the correctness of a possible output can be checked in polynomial time in the input and output.
The definition of a computably enumerable set as the domain of a partial function, rather than the range of a total computable function, is common in contemporary texts. This choice is motivated by the fact that in generalized recursion theories, such as α-recursion theory , the definition corresponding to domains has been found to be more ...
In computer programming, an enumerated type (also called enumeration, enum, or factor in the R programming language, and a categorical variable in statistics) is a data type consisting of a set of named values called elements, members, enumeral, or enumerators of the type.
Sigma function: Sums of powers of divisors of a given natural number. Euler's totient function: Number of numbers coprime to (and not bigger than) a given one. Prime-counting function: Number of primes less than or equal to a given number. Partition function: Order-independent count of ways to write a given positive integer as a sum of positive ...
The geometric distribution, a discrete distribution which describes the number of attempts needed to get the first success in a series of independent Bernoulli trials, or alternatively only the number of losses before the first success (i.e. one less). The Hermite distribution; The logarithmic (series) distribution; The mixed Poisson distribution
Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infinite collection of finite sets S i indexed by the natural numbers, enumerative combinatorics seeks to describe a counting function which counts the number of objects in S n for each n.
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