uppröðun

Uppröðun is the act of arranging objects into a particular order. In Icelandic, the term is commonly used to describe permutations, the mathematical concept of counting and listing all possible orders of a set of items. It can also refer more broadly to organizing data or elements according to a rule or key.

In combinatorics, a permutation is an arrangement where order matters. The number of permutations of n distinct items is n factorial (n!). If some elements are identical, the count is adjusted by dividing by the factorials of the repeated frequencies (for a multiset with frequencies n1, n2, etc., the number of distinct permutations is n!/(n1! n2! ...)).

If arrangements are chosen from a larger set with no repetition, the number of possible ordered sequences of length k is denoted P(n, k) = n!/(n−k)!. If repetition is allowed, the number of length-k sequences from n items is n^k.

In computing and data processing, uppröðun relates to sorting or ordering data by a key. Sorting algorithms produce an ordered sequence; distinctions exist between stable and unstable sorts, with time complexities often around O(n log n) for efficient comparison-based methods.

Example: for the set {A, B, C}, there are 3! = 6 distinct permutations: ABC, ACB, BAC, BCA, CAB, CBA. This illustrates the core idea of uppröðun as the enumeration of all possible orders of elements.