permutacemi

Perumutacemi is the instrumental plural form of permutace in Czech, and in mathematical usage it refers to permutations. In combinatorics, a permutation of a set with n elements is a bijection from the set to itself, equivalently a rearrangement of its elements. The collection of all permutations on n elements forms the symmetric group Sn under composition and has order n!.

A permutation can be described in cycle notation as a product of disjoint cycles. Transpositions, 2-cycles, are basic building blocks, and every permutation can be decomposed into transpositions. The parity of a permutation is determined by the number of transpositions in such a decomposition: even or odd.

Counting and arrangements: If all elements are distinct, there are n! permutations. If some elements repeat, the number of distinct permutations of a multiset is n! divided by the product of the factorials of the multiplicities of each element. Special topics include derangements (permutations with no fixed points) counted by subfactorials and permutations with restricted positions.

Applications and related concepts: Permutations arise in sorting, cryptography, coding theory, experimental design, scheduling, and studies of genome rearrangements in biology. Algorithms to generate permutations include lexicographic ordering and Heap’s algorithm, with typical complexity proportional to the number of permutations, O(n!). Related ideas cover permutation groups, cycle types, and the notion of permutation motifs in various mathematical and applied contexts.

In Czech mathematical literature, permutace and its instrumental plural permutacemi are standard terms; in English, the corresponding term is simply permutations.