# 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",
"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"));

• I have answered another question like this. I hope it will help you also. Please check: stackoverflow.com/a/65535210/2184182 Commented Jan 2, 2021 at 1:34
• Can someone please explain how the single & operator is working here? I can find no explanation for this. Commented Nov 23, 2022 at 17:03

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?

• Stumbled back here and have no idea why I didn't originally point out that "combination" also has an existing mathematical definition en.wikipedia.org/wiki/Combination Commented Nov 10, 2013 at 6:41
• "All possible combinations" can also be called a "power set". Commented Jan 2, 2014 at 17:28
• On closer inspection… a power set should include the empty string. I'm not sure what to call a power set minus the empty string. Commented Jan 21, 2014 at 17:24
• I'm trying to create my library of babel and this came in handy xD
– Kyle
Commented Oct 1, 2015 at 11:36
• Instead of "power set" (which is about sets, without caring for order), an even better name would be subsequences (although technically that includes the empty sequence as well). Commented Jul 31, 2016 at 23:24

You can do it recursively:

function getCombinations(chars) {
var result = [];
var f = function(prefix, chars) {
for (var i = 0; i < chars.length; i++) {
result.push(prefix + chars[i]);
f(prefix + chars[i], chars.slice(i + 1));
}
}
f('', chars);
return result;
}


Usage:

var combinations = getCombinations(["a", "b", "c", "d"]);


Result:

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

• thanks... slight tweak to return set of arrays function getCombinations(array) { var result = []; var f = function(prefix=[], array) { for (var i = 0; i < array.length; i++) { result.push([...prefix,array[i]]); f([...prefix,array[i]], array.slice(i + 1)); } } f('', array); return result; }javascript
– mc.
Commented Jun 3, 2019 at 21:01
• The best and most readable solution! Commented Aug 17, 2019 at 11:59

I prefer your approach much better than a recursive approach, especially when larger lists are being processed.

Some notes:

• I like the name powerSet as per @200_success
• You do not need to check for combination.length !== 0 if you start with i=1
• If you call the function permutations, then you should not call the list you build combinations, that is confusing
• You could cache list.length, that is a common optimization

With curly braces you can then have:

function powerSet( list ){
var set = [],
listSize = list.length,
combinationsCount = (1 << listSize),
combination;

for (var i = 1; i < combinationsCount ; i++ ){
var combination = [];
for (var j=0;j<listSize;j++){
if ((i & (1 << j))){
combination.push(list[j]);
}
}
set.push(combination);
}
return set;
}


without them it could look like this:

function powerSet( list ){
var set = [],
listSize = list.length,
combinationsCount = (1 << listSize);

for (var i = 1; i < combinationsCount ; i++ , set.push(combination) )
for (var j=0, combination = [];j<listSize;j++)
if ((i & (1 << j)))
combination.push(list[j]);
return set;
}

• The first version is excellent. The second one is not nearly as good. Commented Jan 21, 2014 at 17:21
• The 2nd version is too Golfic, but I am addicted to dropping curlies. Commented Jan 21, 2014 at 17:39
• I like this answer. If I were to try to improve (which is not needed), I would say, 1. I would use powerSet, instead of set, or at least something with less overloaded meaning. 2. I would avoid 2nd nested loop as most of the inner loops are unused. I think memoization would be good for that (jsfiddle.net/ynotravid/kkwfnpu7/20).
– Ynot
Commented Apr 10, 2018 at 23:30
• @PirateApp From OP's example, order doesn't matter, he only wants one 4-character output, i.e. either one of abcd, adbc, etc. Commented May 19, 2021 at 10:10
• @blackr1234, yes you are right thanks. jsfiddle.net/ynotravid/0qp6ctzn/4
– Ynot
Commented May 24, 2021 at 0:44

In this case, a non-recursive solution is easier to understand, IMHO:

const powerset = (array) => { // O(2^n)
const results = [[]];
for (const value of array) {
const copy = [...results]; // See note below.
for (const prefix of copy) {
results.push(prefix.concat(value));
}
}
return results;
};

console.log(
powerset(['A', 'B', 'C'])
);

// [ [],
//   [ 'A' ],
//   [ 'B' ],
//   [ 'A', 'B' ],
//   [ 'C' ],
//   [ 'A', 'C' ],
//   [ 'B', 'C' ],
//   [ 'A', 'B', 'C' ] ]

Because results is extended within the loop body, we cannot iterate over it using for-of — doing so would iterate over the newly added elements as well, resulting in an infinite loop. We only want to iterate over the elements that are in results when the loop starts, i.e. indices 0 until results.length - 1. So we either cache the original length in a variable and use that, i.e.

for (let index = 0, length = results.length; index < length; index++) {
const prefix = results[index];
// …
}


…or we just create a static copy of results and iterate over that.

An alternative is to build a trie and then walk the trie to generate the combinations. There are two recursive functions and I've timed it as roughly an order of magnitude slower than your iterative version, but I thought you might find it interesting nonetheless. (Once JS gets tail-call optimisation, some recursive approaches will run faster.)

var follows, combinations;

follows = function(a){
return a.map(function(item, i){
return [item, follows(a.slice(i+1))];
});
};

combinations = function(a){
var combs = function(prefix, trie, result){
trie.forEach(function(node, i){
result.push(prefix + node[0]);
combs(prefix + node[0], node[1], result);
});
return result;
};
return combs('', follows(a), []);
};

combinations(['a','b','c','d']);


P.S. Your permutations function outputs an array of arrays, not an array of strings like your example at the top of your question. I've output an array of strings with my combinations function.

A much faster way to do Math.pow( 2, x ) if x is an integer is 1 << x.

A good name for the function might also be 'array_permutator'.

• My main issue is this double-looping with the if-checking. Commented Dec 19, 2011 at 20:33
• Yes, I understand that. I cannot come up with an idea for eliminating the nested loop. So, wait and see if someone thinks of something. Commented Dec 19, 2011 at 20:39

I have two solutions for this, one being binary and one being recursive;

The binary would be as follows;

function getSubArrays(arr){
var len = arr.length,
subs = Array(Math.pow(2,len)).fill();
return subs.map((_,i) => { var j = -1,
k = i,
res = [];
while (++j < len ) {
k & 1 && res.push(arr[j]);
k = k >> 1;
}
return res;
}).slice(1);
}

console.log(JSON.stringify(getSubArrays(["a","b","c","d"])));

And the recursive one is as follows;

function getSubArrays(arr){
if (arr.length === 1) return [arr];
else {
subarr = getSubArrays(arr.slice(1));
return subarr.concat(subarr.map(e => e.concat(arr[0])), [[arr[0]]]);
}
}

console.log(JSON.stringify(getSubArrays(["a","b","c","d"])));

In this collection of answers, I think we're missing the traditional recursive solution, which highlights the elegance of a recursive algorithm:

const subSeq = (s) => {
if (s.length == 1) return ['', s]

// All the subSeq without the first char:
const ss = subSeq(s.slice(1))

// Return both those with and those without the first char
return [...ss.map(ch => s[0] + ch), ...ss]
}


or more concisely stated, but perhaps harder to follow:

const subSeq = (s) => s.length == 1
? ['', s]
: (ss=subSeq(s.slice(1)),
[...ss.map(ch => s[0] + ch), ...ss])


includes the empty string in the answer:

subSeq('abcd')
=> ["abc", "abcd", "ab", "abd", "ac", "acd", "a", "ad", "bc", "bcd", "b", "bd", "c", "cd", "", "d"]


Try this simple one.

var input="abcde";
var len=input.length;
var str = "";
var p=Math.pow(2,len);
var twoPower;
for(i = 0; i < p; i++)
{
twoPower=p;
for(j=0;j<len;j++)
{
twoPower=twoPower/2;
str+= (i & twoPower ? input.charAt(j) : "");
}
str+="\n";
}


It's working method is so simple. What would you do if you have to find all binary numbers for a given a bit length? Yes the 8 4 2 1 method. if bit length is 3 then possible numbers are

## 4 2 1

0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1

These are the 8 possible values. The same method is used here but for 1's here is a character and for 0's nothing. So, number of possible combinations is (2^n)-1.

• Welcome to CodeReview. As it is your code is little more than a code dump, please provide context towards why the OP should take your suggestion /what would differentiate it from what he's already doing. Commented Apr 1, 2015 at 17:21