## Lexicographic Permutations

### March 9, 2010

Edsger Dijkstra, in his book A Discipline of Programming, describes a function that, given a list of elements and a function that returns a total ordering of those elements, produces the next lexicographic permutation of the list; for instance, the permutations of the list (1 2 3 4) in lexicographic order are (1 2 3 4) (2 1 3 4) (1 3 2 4) (3 1 2 4) (2 3 1 4) (3 2 1 4) (1 2 4 3) (2 1 4 3) (1 4 2 3) (4 1 2 3) (2 4 1 3) (4 2 1 3) (1 3 4 2) (3 1 4 2) (1 4 3 2) (4 1 3 2) (3 4 1 2) (4 3 1 2) (2 3 4 1) (3 2 4 1) (2 4 3 1) (4 2 3 1) (3 4 2 1) (4 3 2 1). With the least significant item in the list at the head, the next greater element than (1 2 3 4) is (2 1 3 4), since the most significant tail of the list doesn’t change, and the next greater permutation differs in the least significant elements. To make a greater permutation, some element of the list must be replaced by a larger element to its left; to make the next permutation, the replacement must happen as far to the left as possible, in the least significant position, and the replacing element must be as small as possible.

Your task is to write functions that return the next lexicographic permutation and a list of all permutations of a list. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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