A New Method for Generating Permutations in Lexicographic Order

Abstract

First, an ordinal representation scheme for permutations is defined. Next, an ”unranking” algorithm that can generate a permutation of n items according to its ordinal representation is designed. By using this algorithm, all permutations can be systematically generated in lexicographic order. Finally, a ”ranking” algorithm that can convert a permutation to its ordinal representation is designed. By using these ”ranking” and ”unranking” algorithms, any permutation that is positioned in lexicographic order, away from a given permutation by any specific distance, can be generated. This significant benefit is the main difference between the proposed method and previously published alternatives. Three other advantages are as follows: First, not being restricted to sequentially numbering the n items from one to n, this method can even handle items with non-numeral marks without the aid of mapping. Second, this approach is well suited to a wide variety of permutation generations such as alternating permutations and derangements. Third, the proposed method can also be extended to a multiset.