Generating all combinations of an array

I'm generating all combinations of an array, so for instance, ["a", "b", "c", "d"] will generate:

[
  "a",    "b",    "ab",   "c",    "ac",
  "bc",   "abc",  "d",    "ad",   "bd",
  "abd",  "cd",   "acd",  "bcd",  "abcd"
]

Here's the code I've written that does complete this task.

What I'd like to know is if there is a better way, as iterating over the array twice feels like I'm cheating, or the complexity of the code is much more computationally expensive than it needs to be.

Also, the name for a function that takes an array and returns the combinations, what might that be called? Combinator seems inappropriate.

var letters = ["a", "b", "c", "d"];
var combi = [];
var temp= "";
var letLen = Math.pow(2, letters.length);

for (var i = 0; i < letLen ; i++){
    temp= "";
    for (var j=0;j<letters.length;j++) {
        if ((i & Math.pow(2,j))){ 
            temp += letters[j]
        }
    }
    if (temp !== "") {
        combi.push(temp);
    }
}

console.log(combi.join("\n"));

A recursive solution, originally seen here, but modified to fit your requirements (and look a little more JavaScript-y):

function combinations(str) {
    var fn = function(active, rest, a) {
        if (!active && !rest)
            return;
        if (!rest) {
            a.push(active);
        } else {
            fn(active + rest[0], rest.slice(1), a);
            fn(active, rest.slice(1), a);
        }
        return a;
    }
    return fn("", str, []);
}

Test:

combinations("abcd")

Output:

["abcd", "abc", "abd", "ab", "acd", "ac", "ad", "a", "bcd", "bc", "bd", "b", "cd", "c", "d"]

Regarding the name: Don't name it permutations; a permutation is an arrangement of all the original elements (of which there should be n! total). In other words, it already has a precise meaning; don't unnecessarily overload it. Why not simply name it combinations?

StackExchange Code Review Q#7001, answer score: 57