pnth

Pnth is a term used in theoretical computer science and mathematics to denote an operator that returns the n-th element of a sequence that satisfies a given predicate. The prefix p stands for a predicate or property, and nth indicates indexing within the filtered subsequence.

Formally, let S be a sequence (s1, s2, s3, ...), p(x) a boolean predicate, and n a positive integer. pnth(S, p, n) is the unique element sk such that p(sk) is true and exactly n−1 elements among s1, ..., sk−1 satisfy p. If there are fewer than n elements in S satisfying p, pnth is undefined or returns a designated sentinel value.

Equivalently, pnth(S, p, n) equals the n-th element of the filtered sequence S|p, where S|p = (si in S : p(si)). In programming, pnth can be implemented by filtering then indexing, or by a streaming algorithm that counts until reaching n. The definition is independent of the underlying data structure as long as the sequence order is preserved.

Applications of pnth appear in formal descriptions of selection under constraints, algorithmic design that involves constrained sampling, and the study of order statistics within filtered data. It is also used in discussions of lazy evaluation and query languages where a predicate defines a restricted domain from which the n-th element is retrieved.

Etymology and usage vary by author; pnth is not a standardized operator in mainstream mathematics or computer science. When used, it is typically clarified by specifying the exact predicate and the indexing convention to avoid ambiguity. See also nth_element, filter, and sequence.