Permutations of Tuple

Problem Link

Problem

Given a generic tuple type T extends unknown[], write a type which produces all permutations of T as a union.

PermutationsOfTuple<[1, 2, 3]>
// [1, 2, 3] | [1, 3, 2] | [2, 1, 3] | [2, 3, 1] | [3, 1, 2] | [3, 2, 1]

Solution

type PermutationsOfTuple<
  T extends unknown[],
  Prefix extends unknown[] = []
> =
  T extends [infer First, ...infer Rest]
    ?
      | PermutationsOfTuple<Rest, [...Prefix, First]>
      | (Rest extends [] ? never : PermutationsOfTuple<[...Rest, First], Prefix>)
    : Prefix

A cleaner alternative using "pick one element" recursion:

type PrependToAll<T, U extends any[][]> =
  U extends [infer First extends any[], ...infer Rest extends any[][]]
    ? [[T, ...First], ...PrependToAll<T, Rest>]
    : []

type PermutationsOfTuple<T extends unknown[]> =
  T extends [infer First, ...infer Rest]
    ? [...PrependToAll<First, PermutationsOfTuple<Rest>>, ...PermutationsOfTuple<Rest extends [] ? never : [...Rest, First]>]
    : [T]

Simplest readable approach:

type PermutationsOfTuple<T extends unknown[], R extends unknown[] = []> =
  T extends [infer F, ...infer L]
    ? PermutationsOfTuple<L, [...R, F]> | PermutationsOfTuple<[...L, F], R>
    : R

How it works:

1. At each step, pick the first element F and either:
   • Include it at the end of the accumulated prefix R, then recurse on L.
   • Move it to the end of remaining L, then recurse — effectively trying all positions.
2. When T is empty, R is a complete permutation — add it to the union.

Key Takeaways

• Generating permutations requires distributing choices across recursive branches (union of recursive calls).
• Rotating elements ([...L, F]) is a classic way to visit all positions without explicit indexing.
• The result is a union of tuples, not a union of elements.