Inversionsobergrenze

Inversionsobergrenze is a term used in combinatorics and sorting theory to denote the maximal possible number of inversions in a sequence of length n, typically considered over all permutations of n distinct elements. An inversion in a permutation is a pair of positions (i, j) with i < j and the value at i is greater than the value at j. The inversion number inv(π) counts how many such pairs occur for a given permutation π.

The inversionsobergrenze for sequences of length n is n(n−1)/2. This bound is achieved by the strictly decreasing permutation, where every pair of positions forms an inversion. The bound is tight: no sequence of length n can have more than n(n−1)/2 inversions when all elements are distinct.

In cases where the sequence may contain repeated values, the maximum number of inversions can be smaller than n(n−1)/2, because equal elements do not contribute inversions (a_i > a_j is false when a_i = a_j). Nevertheless, among sequences with a fixed multiset of values, a strictly decreasing arrangement of the distinct values yields the maximum possible inversions given the constraints.

Applications of the concept include analyses of sorting algorithms, such as bubble sort, where the number of swaps is tied to the number of inversions. The inversionsobergrenze provides a universal upper bound on that work for sequences of length n with distinct elements. In probability and combinatorics, the average number of inversions in a random permutation is n(n−1)/4.

See also: Inversion (combinatorics), Permutation, Sorting algorithms.