Permutations

Problem

https://leetcode.com/problems/permutations/

Given an array nums of distinct integers, return all the possible permutations. You can return the answer in any order.

Example 1:

Input: nums = [1,2,3]
Output: [[1,2,3],[1,3,2],[2,1,3],[2,3,1],[3,1,2],[3,2,1]]

Example 2:

Input: nums = [0,1]
Output: [[0,1],[1,0]]

Example 3:

Input: nums = [1]
Output: [[1]]

Constraints:

• 1 <= nums.length <= 6

• -10 <= nums[i] <= 10

• All the integers of nums are unique.

Pattern

Array, Backtracking

Approaches

Recursive

Explanation

We can generate permutations inductively. Suppose we have all the permutations of nums[1:]. If we prepend nums[1] to these permutations, we now have all permutations of nums that start with nums[1]. Doing this for each \(n\) in nums, we can generate every permutation of nums.

Code

def permutations(nums: list[int]) -> list[list[int]]:
    """Generate all permutations of ``nums``."""
    if len(nums) > 0:
        return [
            [n] + p
            for i, n in enumerate(nums)
            for p in permutations(nums[:i] + nums[i + 1 :])
        ]
    else:
        return [[]]

Test

>>> from permutations__recursive import permutations
>>> permutations([1, 2])
[[1, 2], [2, 1]]
>>> permutations([1, 2, 3])
[[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]]
>>> len(permutations([1, 2, 3, 4]))
24

Complexity

\(n\) is the length of the input array

Measure	Complexity	Notes
Time	\(O(n!)\)	generate all permutations
Auxiliary Space	\(O(1)\)

permutations__recursive.permutations(nums: list[int]) → list[list[int]]

Generate all permutations of nums.