Generating permutations and combinations in lexicographical order

Generating Permutations and Combinations

We consider permutations of {1, 2, . . . , n} in which each integer is given a direction; such permutations are called directed permutations. An integer k in a directed permutation is called mobile of its arrow points to a smaller integer adjacent to it. For example, for → 3 → 2 ← 5 → 4 → 6 → 1 , the integers 3, 5, and 6 are mobile. It follows that 1 can never be mobile since there is no intege...

Lexicographical Order

procedure ari thmetic (a, b, c, op); in t eger a, b, c, op; ¢ o n l m e n t This procedure will perform different order ar i thmetic operations with b and c, put t ing the result in a. The order of the operation is given by op. For op = 1 addit ion is performed. For op = 2 multiplicaLion, repeated addition, is done. Beyond these the operations are non-commutat ive. For op = 3 exponentiat ion, r...

Permutations and Combinations

An Optimal Systolic Algorithm for Generating Permutations in Lexicographic Order

A New Method for Generating Permutations in Lexicographic Order

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...