Introduction

The Johnson–Trotter algorithm is a method for producing all the permutations of a given list in such a way that the transition from one permutation to the next is accomplished by a single transposition of adjacent elements. In this pearl we calculate a loopless version of the algorithm. The main idea is to make use of one of the loopless programs for the generalised boustrophedon product boxall developed in the previous pearl.

A recursive formulation

In the Johnson–Trotter permutation algorithm the transitions for a list of length n of length greater than one are defined recursively in terms of the transitions for a list of length n−1. Label the elements of the list with positions 0 through n−1 and let the list itself be denoted by xs [x]. Begin with a downward run [n−1, n−2,…, 1], where transition i means “interchange the element at position i with the element at position i−1”. The effect is to move x from the last position to the first, resulting in the final permutation [x] xs. For example, the transitions [3, 2, 1] applied to the string “abcd” result in the three permutations “abdc”, “adbc” and “dabc”. Next, suppose the transitions generating the permutations of xs are [j1, j2,…]. Apply the transition j1+1 to the current permutation [x] xs. We have to increase j1 by one because xs is now one position to the right of the “runner” x.