'Permutations: Johnson's' Algorithm.'įor Mathematicians. 'Permutation Generation Methods.' Comput. The word 'permutation' also refers to the act or process of changing the linear order of an ordered set. New York: W. W. Norton, pp. 239-240, 1942. In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements.

Knuth,Īrt of Computer Programming, Vol. 3: Sorting and Searching, 2nd ed. 'Generation of Permutations byĪdjacent Transpositions.' Math. 'Permutations by Interchanges.' Computer J. 'Arrangement Numbers.' In Theīook of Numbers. The permutation which switches elements 1 and 2 and fixes 3 would be written as (2)(143) all describe the same permutation.Īnother notation that explicitly identifies the positions occupied by elements before and after application of a permutation on elements uses a matrix, where the first row is and the second row is the new arrangement. There is a great deal of freedom in picking the representation of a cyclicĭecomposition since (1) the cycles are disjoint and can therefore be specified inĪny order, and (2) any rotation of a given cycle specifies the same cycle (Skienaġ990, p. 20). This is denoted, corresponding to the disjoint permutation cycles (2)Īnd (143). The unordered subsets containing elements are known as the k-subsetsĪ representation of a permutation as a product of permutation cycles is unique (up to the ordering of the cycles). (Uspensky 1937, p. 18), where is a factorial.