Powerset

Powerset_flatMap.swift

extension Array {
	var powerSet: [[Element]] {
		guard !isEmpty else { return [[]] }
		return Array(self[1...]).powerSet.flatMap { [$0, [self[0]] + $0] }
	}
}

print([1,2,3,4].powerSet) // -> [[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3], [4], [1, 4], [2, 4], [1, 2, 4], [3, 4], [1, 3, 4], [2, 3, 4], [1, 2, 3, 4]]

Powerset_map.swift

extension Array {
	var powerSet: [[Element]] {
		guard !isEmpty else { return [[]] }
		let tailPowerSet = Array(self[1...]).powerSet
		return tailPowerSet.map { [self[0]] + $0 } + tailPowerSet
	}
}

print([1,2,3,4].powerSet) // -> [[1, 2, 3, 4], [1, 2, 3], [1, 2, 4], [1, 2], [1, 3, 4], [1, 3], [1, 4], [1], [2, 3, 4], [2, 3], [2, 4], [2], [3, 4], [3], [4], []]

Hello, JadenGeler.
I will use this gist in my private source. I love it.
If you have any requirement for this use, please email me: kyoo@podotree.com
I will apply it shortly if not conflict between your requirement and my spec.
Thanks :)

You're welcome to use it anywhere you'd like!

So cool. If I understand correctly, it is using induction on the length of array.
The base is length zero, an empty array.
The inductive part is support we have the power set of an array of length n, P(An) , we can computer the power set of an array of length (n + 1), P(An+1) , by for each set p of P(An), it would produce two sets, p and p + A[n], that is itself and itself with the new added element, and the set of those all produced sets is the power set of array length (n + 1), i.e. P(An+1).
Moreover, this algorithm not just suitable for array but for any collection. The following code block is my improvement following your idea. I appreciate any information (comment or suggestion) you could give.
PowerSet of Collection

extension Collection {
  public var powerSet: [[Element]] {
    guard let fisrt = self.first else {return [[]]}
    return self.dropFirst().powerSet.flatMap{[$0, [fisrt] + $0]}
  }
}
let s: Set<Int> = [1,2,3]
s.powerSet //[[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]]
let a: Array<Int> = [1,2,3]
a.powerSet //[[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]]

what is complexity of power set algorithm it seems to growth hugely in my case like O(n!) or O(n^k)