Permutationen

Permutationen, or permutations, are mathematical arrangements of objects in a specific order. For a finite set with n distinct elements, a permutation is a reordering of all elements of the set. The set of all such permutations has size n!, the factorial of n.

For partially ordered selections, formulas differ: the number of ways to choose and arrange r items from n distinct objects without repetition is nPr = n!/(n−r)!. If repetition is allowed and order matters, there are n^r possible sequences.

When the object types are not all distinct, the count must account for identical items. For a multiset with total n elements, where element types occur m1, m2, ..., mk times, the number of distinct permutations is n!/(m1! m2! ... mk!). In group-theoretic terms, a permutation can be viewed as a bijection from a set to itself; the set of all permutations on n elements forms the symmetric group S_n. Permutations can be decomposed into cycles, and the order of a permutation is the least common multiple of its cycle lengths.

Permutationen are contrasted with combinations, where the order of selection is irrelevant. Applications include counting possible orderings, scheduling, cryptography, and the analysis of algorithms. Common special topics include derangements (permutations with no fixed points) and permutation representations in abstract algebra.