Combinations

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Task
Combinations
You are encouraged to solve this task according to the task description, using any language you may know.
Given non-negative integers m and n, generate all size m combinations of the integers from 0 to n-1 in sorted order (each combination is sorted and the entire table is sorted).

For example, 3 comb 5 is

0 1 2
0 1 3
0 1 4
0 2 3
0 2 4
0 3 4
1 2 3
1 2 4
1 3 4
2 3 4

If it is more "natural" in your language to start counting from 1 instead of 0 the combinations can be of the integers from 1 to n.

See Also:

The number of samples of size k from n objects.
With combinations and permutations generation tasks.
Order Unimportant Order Important
Without replacement  \binom nk = ^n\operatorname C_k = \frac{n(n-1)\ldots(n-k+1)}{k(k-1)\dots1} ^n\operatorname P_k = n\cdot(n-1)\cdot(n-2)\cdots(n-k+1)
Task: Combinations Task: Permutations
With replacement  \binom {n+k-1}k = ^{n+k-1}\operatorname C_k = {(n+k-1)! \over (n-1)!k!} nk
Task: Combinations with repetitions Task: Permutations with repetitions

Contents

[edit] Ada

with Ada.Text_IO;  use Ada.Text_IO;
 
procedure Test_Combinations is
generic
type Integers is range <>;
package Combinations is
type Combination is array (Positive range <>) of Integers;
procedure First (X : in out Combination);
procedure Next (X : in out Combination);
procedure Put (X : Combination);
end Combinations;
 
package body Combinations is
procedure First (X : in out Combination) is
begin
X (1) := Integers'First;
for I in 2..X'Last loop
X (I) := X (I - 1) + 1;
end loop;
end First;
procedure Next (X : in out Combination) is
begin
for I in reverse X'Range loop
if X (I) < Integers'Val (Integers'Pos (Integers'Last) - X'Last + I) then
X (I) := X (I) + 1;
for J in I + 1..X'Last loop
X (J) := X (J - 1) + 1;
end loop;
return;
end if;
end loop;
raise Constraint_Error;
end Next;
procedure Put (X : Combination) is
begin
for I in X'Range loop
Put (Integers'Image (X (I)));
end loop;
end Put;
end Combinations;
 
type Five is range 0..4;
package Fives is new Combinations (Five);
use Fives;
 
X : Combination (1..3);
begin
First (X);
loop
Put (X); New_Line;
Next (X);
end loop;
exception
when Constraint_Error =>
null;
end Test_Combinations;

The solution is generic the formal parameter is the integer type to make combinations of. The type range determines n. In the example it is

type Five is range 0..4;

The parameter m is the object's constraint. When n < m the procedure First (selects the first combination) will propagate Constraint_Error. The procedure Next selects the next combination. Constraint_Error is propagated when it is the last one. Sample output:

 0 1 2
 0 1 3
 0 1 4
 0 2 3
 0 2 4
 0 3 4
 1 2 3
 1 2 4
 1 3 4
 2 3 4

[edit] ALGOL 68

Translation of: Python
Works with: ALGOL 68 version Revision 1 - one minor extension to language used - PRAGMA READ, similar to C's #include directive.
Works with: ALGOL 68G version Any - tested with release algol68g-2.6.
File: prelude_combinations.a68
# -*- coding: utf-8 -*- #
 
COMMENT REQUIRED BY "prelude_combinations_generative.a68"
MODE COMBDATA = ~;
PROVIDES:
# COMBDATA*=~* #
# comb*=~ list* #
END COMMENT
 
MODE COMBDATALIST = REF[]COMBDATA;
MODE COMBDATALISTYIELD = PROC(COMBDATALIST)VOID;
 
PROC comb gen combinations = (INT m, COMBDATALIST list, COMBDATALISTYIELD yield)VOID:(
CASE m IN
# case 1: transpose list #
FOR i TO UPB list DO yield(list[i]) OD
OUT
[m + LWB list - 1]COMBDATA out;
INT index out := 1;
FOR i TO UPB list DO
COMBDATA first = list[i];
# FOR COMBDATALIST sub recombination IN # comb gen combinations(m - 1, list[i+1:] #) DO (#,
## (COMBDATALIST sub recombination)VOID:(
out[LWB list ] := first;
out[LWB list+1:] := sub recombination;
yield(out)
# OD #))
OD
ESAC
);
 
SKIP
File: test_combinations.a68
#!/usr/bin/a68g --script #
# -*- coding: utf-8 -*- #
 
CO REQUIRED BY "prelude_combinations.a68" CO
MODE COMBDATA = INT;
#PROVIDES:#
# COMBDATA~=INT~ #
# comb ~=int list ~#
PR READ "prelude_combinations.a68" PR;
 
FORMAT data fmt = $g(0)$;
 
main:(
INT m = 3;
FORMAT list fmt = $"("n(m-1)(f(data fmt)",")f(data fmt)")"$;
FLEX[0]COMBDATA test data list := (1,2,3,4,5);
# FOR COMBDATALIST recombination data IN # comb gen combinations(m, test data list #) DO (#,
## (COMBDATALIST recombination)VOID:(
printf ((list fmt, recombination, $l$))
# OD # ))
)
 
Output:
(1,2,3)
(1,2,4)
(1,2,5)
(1,3,4)
(1,3,5)
(1,4,5)
(2,3,4)
(2,3,5)
(2,4,5)
(3,4,5)

[edit] AppleScript

on comb(n, k)
set c to {}
repeat with i from 1 to k
set end of c to i's contents
end repeat
set r to {c's contents}
repeat while my next_comb(c, k, n)
set end of r to c's contents
end repeat
return r
end comb
 
on next_comb(c, k, n)
set i to k
set c's item i to (c's item i) + 1
repeat while (i > 1 and c's item i ≥ n - k + 1 + i)
set i to i - 1
set c's item i to (c's item i) + 1
end repeat
if (c's item 1 > n - k + 1) then return false
repeat with i from i + 1 to k
set c's item i to (c's item (i - 1)) + 1
end repeat
return true
end next_comb
 
return comb(5, 3)
Output:
{{1, 2, 3}, {1, 2, 4}, {1, 2, 5}, {1, 3, 4}, {1, 3, 5}, {1, 4, 5}, {2, 3, 4}, {2, 3, 5}, {2, 4, 5}, {3, 4, 5}}


[edit] AutoHotkey

contributed by Laszlo on the ahk forum

MsgBox % Comb(1,1)
MsgBox % Comb(3,3)
MsgBox % Comb(3,2)
MsgBox % Comb(2,3)
MsgBox % Comb(5,3)
 
Comb(n,t) { ; Generate all n choose t combinations of 1..n, lexicographically
IfLess n,%t%, Return
Loop %t%
c%A_Index% := A_Index
i := t+1, c%i% := n+1
 
Loop {
Loop %t%
i := t+1-A_Index, c .= c%i% " "
c .= "`n" ; combinations in new lines
j := 1, i := 2
Loop
If (c%j%+1 = c%i%)
c%j% := j, ++j, ++i
Else Break
If (j > t)
Return c
c%j% += 1
}
}

[edit] AWK

BEGIN {
## Default values for r and n (Choose 3 from pool of 5). Can
## alternatively be set on the command line:-
## awk -v r=<number of items being chosen> -v n=<how many to choose from> -f <scriptname>
if (length(r) == 0) r = 3
if (length(n) == 0) n = 5
 
for (i=1; i <= r; i++) { ## First combination of items:
A[i] = i
if (i < r ) printf i OFS
else print i}
 
## While 1st item is less than its maximum permitted value...
while (A[1] < n - r + 1) {
## loop backwards through all items in the previous
## combination of items until an item is found that is
## less than its maximum permitted value:
for (i = r; i >= 1; i--) {
## If the equivalently positioned item in the
## previous combination of items is less than its
## maximum permitted value...
if (A[i] < n - r + i) {
## increment the current item by 1:
A[i]++
## Save the current position-index for use
## outside this "for" loop:
p = i
break}}
## Put consecutive numbers in the remainder of the array,
## counting up from position-index p.
for (i = p + 1; i <= r; i++) A[i] = A[i - 1] + 1
 
## Print the current combination of items:
for (i=1; i <= r; i++) {
if (i < r) printf A[i] OFS
else print A[i]}}
exit}

Usage:

awk -v r=3 -v n=5 -f combn.awk

Output:

1 2 3
1 2 4
1 2 5
1 3 4
1 3 5
1 4 5
2 3 4
2 3 5
2 4 5
3 4 5

[edit] BBC BASIC

      INSTALL @lib$+"SORTLIB"
sort% = FN_sortinit(0,0)
 
M% = 3
N% = 5
 
C% = FNfact(N%)/(FNfact(M%)*FNfact(N%-M%))
DIM s$(C%)
PROCcomb(M%, N%, s$())
 
CALL sort%, s$(0)
FOR I% = 0 TO C%-1
PRINT s$(I%)
NEXT
END
 
DEF PROCcomb(C%, N%, s$())
LOCAL I%, U%
FOR U% = 0 TO 2^N%-1
IF FNbits(U%) = C% THEN
s$(I%) = FNlist(U%)
I% += 1
ENDIF
NEXT
ENDPROC
 
DEF FNbits(U%)
LOCAL N%
WHILE U%
N% += 1
U% = U% AND (U%-1)
ENDWHILE
= N%
 
DEF FNlist(U%)
LOCAL N%, s$
WHILE U%
IF U% AND 1 s$ += STR$(N%) + " "
N% += 1
U% = U% >> 1
ENDWHILE
= s$
 
DEF FNfact(N%)
IF N%<=1 THEN = 1 ELSE = N%*FNfact(N%-1)
 

[edit] Bracmat

The program first constructs a pattern with m variables and an expression that evaluates m variables into a combination. Then the program constructs a list of the integers 0 ... n-1. The real work is done in the expression !list:!pat. When a combination is found, it is added to the list of combinations. Then we force the program to backtrack and find the next combination by evaluating the always failing ~. When all combinations are found, the pattern fails and we are in the rhs of the last | operator.

(comb=
bvar combination combinations list m n pat pvar var
.  !arg:(?m.?n)
& ( pat
=  ?
& !combinations (.!combination):?combinations
& ~
)
& :?list:?combination:?combinations
& whl
' ( !m+-1:~<0:?m
& chu$(utf$a+!m):?var
& glf$('(%@?.$var)):(=?pvar)
& '(? ()$pvar ()$pat):(=?pat)
& glf$('(!.$var)):(=?bvar)
& ( '$combination:(=)
& '$bvar:(=?combination)
| '($bvar ()$combination):(=?combination)
)
)
& whl
' (!n+-1:~<0:?n&!n !list:?list)
& !list:!pat
| !combinations);
comb$(3.5)  
(.0 1 2)
(.0 1 3)
(.0 1 4)
(.0 2 3)
(.0 2 4)
(.0 3 4)
(.1 2 3)
(.1 2 4)
(.1 3 4)
(.2 3 4)

[edit] C

#include <stdio.h>
 
/* Type marker stick: using bits to indicate what's chosen. The stick can't
* handle more than 32 items, but the idea is there; at worst, use array instead */

typedef unsigned long marker;
marker one = 1;
 
void comb(int pool, int need, marker chosen, int at)
{
if (pool < need + at) return; /* not enough bits left */
 
if (!need) {
/* got all we needed; print the thing. if other actions are
* desired, we could have passed in a callback function. */

for (at = 0; at < pool; at++)
if (chosen & (one << at)) printf("%d ", at);
printf("\n");
return;
}
/* if we choose the current item, "or" (|) the bit to mark it so. */
comb(pool, need - 1, chosen | (one << at), at + 1);
comb(pool, need, chosen, at + 1); /* or don't choose it, go to next */
}
 
int main()
{
comb(5, 3, 0, 0);
return 0;
}

[edit] Lexicographic ordered generation

Without recursions, generate all combinations in sequence. Basic logic: put n items in the first n of m slots; each step, if right most slot can be moved one slot further right, do so; otherwise find right most item that can be moved, move it one step and put all items already to its right next to it.

#include <stdio.h>
 
void comb(int m, int n, unsigned char *c)
{
int i;
for (i = 0; i < n; i++) c[i] = n - i;
 
while (1) {
for (i = n; i--;)
printf("%d%c", c[i], i ? ' ': '\n');
 
/* this check is not strictly necessary, but if m is not close to n,
it makes the whole thing quite a bit faster */

if (c[i]++ < m) continue;
 
for (i = 0; c[i] >= m - i;) if (++i >= n) return;
for (c[i]++; i; i--) c[i-1] = c[i] + 1;
}
}
 
int main()
{
unsigned char buf[100];
comb(5, 3, buf);
return 0;
}

[edit] C++

#include <algorithm>
#include <iostream>
#include <string>
 
void comb(int N, int K)
{
std::string bitmask(K, 1); // K leading 1's
bitmask.resize(N, 0); // N-K trailing 0's
 
// print integers and permute bitmask
do {
for (int i = 0; i < N; ++i) // [0..N-1] integers
{
if (bitmask[i]) std::cout << " " << i;
}
std::cout << std::endl;
} while (std::prev_permutation(bitmask.begin(), bitmask.end()));
}
 
int main()
{
comb(5, 3);
}

Output:

 0 1 2
 0 1 3
 0 1 4
 0 2 3
 0 2 4
 0 3 4
 1 2 3
 1 2 4
 1 3 4
 2 3 4

[edit] C#

using System;
using System.Collections.Generic;
 
public class Program
{
public static IEnumerable<int[]> Combinations(int m, int n)
{
int[] result = new int[m];
Stack<int> stack = new Stack<int>();
stack.Push(0);
 
while (stack.Count > 0) {
int index = stack.Count - 1;
int value = stack.Pop();
 
while (value < n) {
result[index++] = value++;
stack.Push(value);
if (index == m) {
yield return result;
break;
}
}
}
}
 
static void Main()
{
foreach (int[] c in Combinations(3, 5))
{
for (int i = 0; i < c.Length; i++)
{
Console.Write(c[i] + " ");
}
Console.WriteLine();
}
}
}

Here is another implementation that uses recursion, intead of an explicit stack:

 
using System;
using System.Collections.Generic;
 
public class Program
{
public static IEnumerable<int[]> FindCombosRec(int[] buffer, int done, int begin, int end)
{
for (int i = begin; i < end; i++)
{
buffer[done] = i;
 
if (done == buffer.Length - 1)
yield return buffer;
else
foreach (int[] child in FindCombosRec(buffer, done+1, i+1, end))
yield return child;
}
}
 
public static IEnumerable<int[]> FindCombinations(int m, int n)
{
return FindCombosRec(new int[m], 0, 0, n);
}
 
static void Main()
{
foreach (int[] c in FindCombinations(3, 5))
{
for (int i = 0; i < c.Length; i++)
{
Console.Write(c[i] + " ");
}
Console.WriteLine();
}
}
}
 

[edit] Clojure

(defn combinations
"If m=1, generate a nested list of numbers [0,n)
If m>1, for each x in [0,n), and for each list in the recursion on [x+1,n), cons the two"

[m n]
(letfn [(comb-aux
[m start]
(if (= 1 m)
(for [x (range start n)]
(list x))
(for [x (range start n)
xs (comb-aux (dec m) (inc x))]
(cons x xs))))]
(comb-aux m 0)))
 
(defn print-combinations
[m n]
(doseq [line (combinations m n)]
(doseq [n line]
(printf "%s " n))
(printf "%n")))

[edit] CoffeeScript

Basic backtracking solution.

 
combinations = (n, p) ->
return [ [] ] if p == 0
i = 0
combos = []
combo = []
while combo.length < p
if i < n
combo.push i
i += 1
else
break if combo.length == 0
i = combo.pop() + 1
 
if combo.length == p
combos.push clone combo
i = combo.pop() + 1
combos
 
clone = (arr) -> (n for n in arr)
 
N = 5
for i in [0..N]
console.log "------ #{N} #{i}"
for combo in combinations N, i
console.log combo
 
 

output

> coffee combo.coffee 
------ 5 0
[]
------ 5 1
[ 0 ]
[ 1 ]
[ 2 ]
[ 3 ]
[ 4 ]
------ 5 2
[ 0, 1 ]
[ 0, 2 ]
[ 0, 3 ]
[ 0, 4 ]
[ 1, 2 ]
[ 1, 3 ]
[ 1, 4 ]
[ 2, 3 ]
[ 2, 4 ]
[ 3, 4 ]
------ 5 3
[ 0, 1, 2 ]
[ 0, 1, 3 ]
[ 0, 1, 4 ]
[ 0, 2, 3 ]
[ 0, 2, 4 ]
[ 0, 3, 4 ]
[ 1, 2, 3 ]
[ 1, 2, 4 ]
[ 1, 3, 4 ]
[ 2, 3, 4 ]
------ 5 4
[ 0, 1, 2, 3 ]
[ 0, 1, 2, 4 ]
[ 0, 1, 3, 4 ]
[ 0, 2, 3, 4 ]
[ 1, 2, 3, 4 ]
------ 5 5
[ 0, 1, 2, 3, 4 ]
 
 


[edit] Common Lisp

(defun map-combinations (m n fn)
"Call fn with each m combination of the integers from 0 to n-1 as a list. The list may be destroyed after fn returns."
(let ((combination (make-list m)))
(labels ((up-from (low)
(let ((start (1- low)))
(lambda () (incf start))))
(mc (curr left needed comb-tail)
(cond
((zerop needed)
(funcall fn combination))
((= left needed)
(map-into comb-tail (up-from curr))
(funcall fn combination))
(t
(setf (first comb-tail) curr)
(mc (1+ curr) (1- left) (1- needed) (rest comb-tail))
(mc (1+ curr) (1- left) needed comb)))))
(mc 0 n m combination))))

Example use

> (map-combinations 3 5 'print)

(0 1 2) 
(0 1 3) 
(0 1 4) 
(0 2 3) 
(0 2 4) 
(0 3 4) 
(1 2 3) 
(1 2 4) 
(1 3 4) 
(2 3 4) 
(2 3 4)

[edit] Recursive method

(defun comb (m list fn)
(labels ((comb1 (l c m)
(when (>= (length l) m)
(if (zerop m) (return-from comb1 (funcall fn c)))
(comb1 (cdr l) c m)
(comb1 (cdr l) (cons (first l) c) (1- m)))))
(comb1 list nil m)))
 
(comb 3 '(0 1 2 3 4 5) #'print)

[edit] Alternate, iterative method

(defun next-combination (n a)
(let ((k (length a)) m)
(loop for i from 1 do
(when (> i k) (return nil))
(when (< (aref a (- k i)) (- n i))
(setf m (aref a (- k i)))
(loop for j from i downto 1 do
(incf m)
(setf (aref a (- k j)) m))
(return t)))))
 
(defun all-combinations (n k)
(if (or (< k 0) (< n k)) '()
(let ((a (make-array k)))
(loop for i below k do (setf (aref a i) i))
(loop collect (coerce a 'list) while (next-combination n a)))))
 
(defun map-combinations (n k fun)
(if (and (>= k 0) (>= n k))
(let ((a (make-array k)))
(loop for i below k do (setf (aref a i) i))
(loop do (funcall fun (coerce a 'list)) while (next-combination n a)))))
 
; all-combinations returns a list of lists
 
> (all-combinations 4 3)
((0 1 2) (0 1 3) (0 2 3) (1 2 3))
 
; map-combinations applies a function to each combination
 
> (map-combinations 6 4 #'print)
(0 1 2 3)
(0 1 2 4)
(0 1 2 5)
(0 1 3 4)
(0 1 3 5)
(0 1 4 5)
(0 2 3 4)
(0 2 3 5)
(0 2 4 5)
(0 3 4 5)
(1 2 3 4)
(1 2 3 5)
(1 2 4 5)
(1 3 4 5)
(2 3 4 5)

[edit] D

[edit] Slow Recursive Version

Translation of: Python
T[][] comb(T)(in T[] arr, in int k) pure nothrow {
if (k == 0) return [[]];
typeof(return) result;
foreach (immutable i, immutable x; arr)
foreach (suffix; arr[i + 1 .. $].comb(k - 1))
result ~= x ~ suffix;
return result;
}
 
void main() {
import std.stdio;
[0, 1, 2, 3].comb(2).writeln;
}
Output:
[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]]

[edit] More Functional Recursive Version

Translation of: Haskell

Same output.

import std.stdio, std.algorithm, std.range;
 
immutable(int)[][] comb(immutable int[] s, in int m) pure nothrow @safe {
if (!m) return [[]];
if (s.empty) return [];
return s[1 .. $].comb(m - 1).map!(x => s[0] ~ x).array ~ s[1 .. $].comb(m);
}
 
void main() {
4.iota.array.comb(2).writeln;
}

[edit] Lazy Version

module combinations3;
 
import std.traits: Unqual;
 
struct Combinations(T, bool copy=true) {
Unqual!T[] pool, front;
size_t r, n;
bool empty = false;
size_t[] indices;
size_t len;
bool lenComputed = false;
 
this(T[] pool_, in size_t r_) pure nothrow @safe {
this.pool = pool_.dup;
this.r = r_;
this.n = pool.length;
if (r > n)
empty = true;
indices.length = r;
foreach (immutable i, ref ini; indices)
ini = i;
front.length = r;
foreach (immutable i, immutable idx; indices)
front[i] = pool[idx];
}
 
@property size_t length() /*logic_const*/ pure nothrow @nogc {
static size_t binomial(size_t n, size_t k) pure nothrow @safe @nogc
in {
assert(n > 0, "binomial: n must be > 0.");
} body {
if (k < 0 || k > n)
return 0;
if (k > (n / 2))
k = n - k;
size_t result = 1;
foreach (size_t d; 1 .. k + 1) {
result *= n;
n--;
result /= d;
}
return result;
}
 
if (!lenComputed) {
// Set cache.
len = binomial(n, r);
lenComputed = true;
}
return len;
}
 
void popFront() pure nothrow @safe {
if (!empty) {
bool broken = false;
size_t pos = 0;
foreach_reverse (immutable i; 0 .. r) {
pos = i;
if (indices[i] != i + n - r) {
broken = true;
break;
}
}
if (!broken) {
empty = true;
return;
}
indices[pos]++;
foreach (immutable j; pos + 1 .. r)
indices[j] = indices[j - 1] + 1;
static if (copy)
front = new Unqual!T[front.length];
foreach (immutable i, immutable idx; indices)
front[i] = pool[idx];
}
}
}
 
Combinations!(T, copy) combinations(bool copy=true, T)
(T[] items, in size_t k)
in {
assert(items.length, "combinations: items can't be empty.");
} body {
return typeof(return)(items, k);
}
 
// Compile with -version=combinations3_main to run main.
version(combinations3_main)
void main() {
import std.stdio, std.array, std.algorithm;
[1, 2, 3, 4].combinations!false(2).array.writeln;
[1, 2, 3, 4].combinations!true(2).array.writeln;
[1, 2, 3, 4].combinations(2).map!(x => x).writeln;
}

[edit] Lazy Lexicographical Combinations

Includes an algorithm to find mth Lexicographical Element of a Combination.

module combinations4;
import std.stdio, std.algorithm, std.conv;
 
ulong choose(int n, int k) nothrow
in {
assert(n >= 0 && k >= 0, "choose: no negative input.");
} body {
static ulong[][] cache;
 
if (n < k)
return 0;
else if (n == k)
return 1;
while (n >= cache.length)
cache ~= [1UL]; // = choose(m, 0);
auto kmax = min(k, n - k);
while(kmax >= cache[n].length) {
immutable h = cache[n].length;
cache[n] ~= choose(n - 1, h - 1) + choose(n - 1, h);
}
 
return cache[n][kmax];
}
 
int largestV(in int p, in int q, in long r) nothrow
in {
assert(p > 0 && q >= 0 && r >= 0, "largestV: no negative input.");
} body {
auto v = p - 1;
while (choose(v, q) > r)
v--;
return v;
}
 
struct Comb {
immutable int n, m;
 
@property size_t length() const /*nothrow*/ {
return to!size_t(choose(n, m));
}
 
int[] opIndex(in size_t idx) const {
if (m < 0 || n < 0)
return [];
if (idx >= length)
throw new Exception("Out of bound");
ulong x = choose(n, m) - 1 - idx;
int a = n, b = m;
auto res = new int[m];
foreach (i; 0 .. m) {
a = largestV(a, b, x);
x = x - choose(a, b);
b = b - 1;
res[i] = n - 1 - a;
}
return res;
}
 
int opApply(int delegate(ref int[]) dg) const {
int[] yield;
 
foreach (i; 0 .. length) {
yield = this[i];
if (dg(yield))
break;
}
 
return 0;
}
 
static auto On(T)(in T[] arr, in int m) {
auto comb = Comb(arr.length, m);
 
return new class {
@property size_t length() const /*nothrow*/ {
return comb.length;
}
 
int opApply(int delegate(ref T[]) dg) const {
auto yield = new T[m];
 
foreach (c; comb) {
foreach (idx; 0 .. m)
yield[idx] = arr[c[idx]];
if (dg(yield))
break;
}
 
return 0;
}
};
}
}
 
 
version(combinations4_main)
void main() {
foreach (c; Comb.On([1, 2, 3], 2))
writeln(c);
}

[edit] E

def combinations(m, range) {
return if (m <=> 0) { [[]] } else {
def combGenerator {
to iterate(f) {
for i in range {
for suffix in combinations(m.previous(), range & (int > i)) {
f(null, [i] + suffix)
}
}
}
}
}
}
? for x in combinations(3, 0..4) { println(x) }

[edit] Egison

 
(define $comb
(lambda [$n $xs]
(match-all xs (list integer)
[(loop $i [1 ,n] <join _ <cons $a_i ...>> _) a])))
 
(test (comb 3 (between 0 4)))
 

Output:

 
{[|0 1 2|] [|0 1 3|] [|0 2 3|] [|1 2 3|] [|0 1 4|] [|0 2 4|] [|0 3 4|] [|1 2 4|] [|1 3 4|] [|2 3 4|]}
 

[edit] Erlang

 
-module(comb).
-compile(export_all).
 
comb(0,_) ->
[[]];
comb(_,[]) ->
[];
comb(N,[H|T]) ->
[[H|L] || L <- comb(N-1,T)]++comb(N,T).
 

[edit] Elena

#define system.
#define system'routines.
#define extensions.
#define extensions'routines.
 
#symbol(const)M = 3.
#symbol(const)N = 5.
 
// --- Numbers ---
 
#symbol numbers = (:anN)
[
arrayControl new &length:(anN int) &each: anIndex [ Integer new:(anIndex + 1) ]
].
 
// --- Program ---
 
#symbol program =
[
#var aNumbers := numbers:N.
control for:(Combinator new:(arrayControl new &length:M &each: i [ aNumbers ])) &do: aRow
[
consoleEx writeLine:aRow.
].
].

[edit] Factor

USING: math.combinatorics prettyprint ;
 
5 iota 3 all-combinations .
{
    { 0 1 2 }
    { 0 1 3 }
    { 0 1 4 }
    { 0 2 3 }
    { 0 2 4 }
    { 0 3 4 }
    { 1 2 3 }
    { 1 2 4 }
    { 1 3 4 }
    { 2 3 4 }
}

This works with any kind of sequence:

{ "a" "b" "c" } 2 all-combinations .
{ { "a" "b" } { "a" "c" } { "b" "c" } }

[edit] Fortran

program Combinations
use iso_fortran_env
implicit none
 
type comb_result
integer, dimension(:), allocatable :: combs
end type comb_result
 
type(comb_result), dimension(:), pointer :: r
integer :: i, j
 
call comb(5, 3, r)
do i = 0, choose(5, 3) - 1
do j = 2, 0, -1
write(*, "(I4, ' ')", advance="no") r(i)%combs(j)
end do
deallocate(r(i)%combs)
write(*,*) ""
end do
deallocate(r)
 
contains
 
function choose(n, k, err)
integer :: choose
integer, intent(in) :: n, k
integer, optional, intent(out) :: err
 
integer :: imax, i, imin, ie
 
ie = 0
if ( (n < 0 ) .or. (k < 0 ) ) then
write(ERROR_UNIT, *) "negative in choose"
choose = 0
ie = 1
else
if ( n < k ) then
choose = 0
else if ( n == k ) then
choose = 1
else
imax = max(k, n-k)
imin = min(k, n-k)
choose = 1
do i = imax+1, n
choose = choose * i
end do
do i = 2, imin
choose = choose / i
end do
end if
end if
if ( present(err) ) err = ie
end function choose
 
subroutine comb(n, k, co)
integer, intent(in) :: n, k
type(comb_result), dimension(:), pointer, intent(out) :: co
 
integer :: i, j, s, ix, kx, hm, t
integer :: err
 
hm = choose(n, k, err)
if ( err /= 0 ) then
nullify(co)
return
end if
 
allocate(co(0:hm-1))
do i = 0, hm-1
allocate(co(i)%combs(0:k-1))
end do
do i = 0, hm-1
ix = i; kx = k
do s = 0, n-1
if ( kx == 0 ) exit
t = choose(n-(s+1), kx-1)
if ( ix < t ) then
co(i)%combs(kx-1) = s
kx = kx - 1
else
ix = ix - t
end if
end do
end do
 
end subroutine comb
 
end program Combinations

Alternatively:

program combinations
 
implicit none
integer, parameter :: m_max = 3
integer, parameter :: n_max = 5
integer, dimension (m_max) :: comb
character (*), parameter :: fmt = '(i0' // repeat (', 1x, i0', m_max - 1) // ')'
 
call gen (1)
 
contains
 
recursive subroutine gen (m)
 
implicit none
integer, intent (in) :: m
integer :: n
 
if (m > m_max) then
write (*, fmt) comb
else
do n = 1, n_max
if ((m == 1) .or. (n > comb (m - 1))) then
comb (m) = n
call gen (m + 1)
end if
end do
end if
 
end subroutine gen
 
end program combinations

Output:

1 2 3
1 2 4
1 2 5
1 3 4
1 3 5
1 4 5
2 3 4
2 3 5
2 4 5
3 4 5

[edit] GAP

# Built-in
Combinations([1 .. n], m);
 
Combinations([1 .. 5], 3);
# [ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 2, 5 ], [ 1, 3, 4 ], [ 1, 3, 5 ],
# [ 1, 4, 5 ], [ 2, 3, 4 ], [ 2, 3, 5 ], [ 2, 4, 5 ], [ 3, 4, 5 ] ]

[edit] Go

package main
 
import (
"fmt"
)
 
func main() {
comb(5, 3, func(c []int) {
fmt.Println(c)
})
}
 
func comb(n, m int, emit func([]int)) {
s := make([]int, m)
last := m - 1
var rc func(int, int)
rc = func(i, next int) {
for j := next; j < n; j++ {
s[i] = j
if i == last {
emit(s)
} else {
rc(i+1, j+1)
}
}
return
}
rc(0, 0)
}

Output:

[0 1 2]
[0 1 3]
[0 1 4]
[0 2 3]
[0 2 4]
[0 3 4]
[1 2 3]
[1 2 4]
[1 3 4]
[2 3 4]

[edit] Groovy

Following the spirit of the Haskell solution.

[edit] In General

A recursive closure must be pre-declared.

def comb
comb = { m, list ->
def n = list.size()
m == 0 ?
[[]] :
(0..(n-m)).inject([]) { newlist, k ->
def sublist = (k+1 == n) ? [] : list[(k+1)..<n]
newlist += comb(m-1, sublist).collect { [list[k]] + it }
}
}

Test program:

def csny = [ "Crosby", "Stills", "Nash", "Young" ]
println "Choose from ${csny}"
(0..(csny.size())).each { i -> println "Choose ${i}:"; comb(i, csny).each { println it }; println() }

Output:

Choose from [Crosby, Stills, Nash, Young]
Choose 0:
[]

Choose 1:
[Crosby]
[Stills]
[Nash]
[Young]

Choose 2:
[Crosby, Stills]
[Crosby, Nash]
[Crosby, Young]
[Stills, Nash]
[Stills, Young]
[Nash, Young]

Choose 3:
[Crosby, Stills, Nash]
[Crosby, Stills, Young]
[Crosby, Nash, Young]
[Stills, Nash, Young]

Choose 4:
[Crosby, Stills, Nash, Young]

[edit] Zero-based Integers

def comb0 = { m, n -> comb(m, (0..<n)) }

Test program:

println "Choose out of 5 (zero-based):"
(0..3).each { i -> println "Choose ${i}:"; comb0(i, 5).each { println it }; println() }

Output:

Choose out of 5 (zero-based):
Choose 0:
[]

Choose 1:
[0]
[1]
[2]
[3]
[4]

Choose 2:
[0, 1]
[0, 2]
[0, 3]
[0, 4]
[1, 2]
[1, 3]
[1, 4]
[2, 3]
[2, 4]
[3, 4]

Choose 3:
[0, 1, 2]
[0, 1, 3]
[0, 1, 4]
[0, 2, 3]
[0, 2, 4]
[0, 3, 4]
[1, 2, 3]
[1, 2, 4]
[1, 3, 4]
[2, 3, 4]

[edit] One-based Integers

def comb1 = { m, n -> comb(m, (1..n)) }

Test program:

println "Choose out of 5 (one-based):"
(0..3).each { i -> println "Choose ${i}:"; comb1(i, 5).each { println it }; println() }

Output:

Choose out of 5 (one-based):
Choose 0:
[]

Choose 1:
[1]
[2]
[3]
[4]
[5]

Choose 2:
[1, 2]
[1, 3]
[1, 4]
[1, 5]
[2, 3]
[2, 4]
[2, 5]
[3, 4]
[3, 5]
[4, 5]

Choose 3:
[1, 2, 3]
[1, 2, 4]
[1, 2, 5]
[1, 3, 4]
[1, 3, 5]
[1, 4, 5]
[2, 3, 4]
[2, 3, 5]
[2, 4, 5]
[3, 4, 5]

[edit] Haskell

It's more natural to extend the task to all (ordered) sublists of size m of a list.

Straightforward, unoptimized implementation with divide-and-conquer:

comb :: Int -> [a] -> [[a]]
comb 0 _ = [[]]
comb _ [] = []
comb m (x:xs) = map (x:) (comb (m-1) xs) ++ comb m xs

In the induction step, either x is not in the result and the recursion proceeds with the rest of the list xs, or it is in the result and then we only need m-1 elements.

Shorter version of the above:

import Data.List (tails)
 
comb :: Int -> [a] -> [[a]]
comb 0 _ = [[]]
comb m l = [x:ys | x:xs <- tails l, ys <- comb (m-1) xs]

To generate combinations of integers between 0 and n-1, use

comb0 m n = comb m [0..n-1]

Similar, for integers between 1 and n, use

comb1 m n = comb m [1..n]

Another method is to use the built in Data.List.subsequences function, filter for subsequences of length m and then sort:

import Data.List (sort, subsequences)
comb m n = sort . filter ((==m) . length) $ subsequences [0..n-1]

And yet another way is to use the list monad to generate all possible subsets:

comb m n = filter ((==m . length) $ filterM (const [True, False]) [0..n-1]

[edit] Dynamic Programming

The first solution is inefficient because it repeatedly calculates the same subproblem in different branches of recursion. For example, comb m (x1:x2:xs) involves computing comb (m-1) (x2:xs) and comb m (x2:xs), both of which (separately) compute comb (m-1) xs. To avoid repeated computation, we can use dynamic programming:

comb :: Int -> [a] -> [[a]]
comb m xs = combsBySize xs !! m
where
combsBySize = foldr f ([[]] : repeat [])
f x next = zipWith (++) (map (map (x:)) ([]:next)) next

[edit] Icon and Unicon

procedure main()
return combinations(3,5,0)
end
 
procedure combinations(m,n,z) # demonstrate combinations
/z := 1
 
write(m," combinations of ",n," integers starting from ",z)
every put(L := [], z to n - 1 + z by 1) # generate list of n items from z
write("Intial list\n",list2string(L))
write("Combinations:")
every write(list2string(lcomb(L,m)))
end
 
procedure list2string(L) # helper function
every (s := "[") ||:= " " || (!L|"]")
return s
end
 
link lists
The provides the core procedure lcomb in lists written by Ralph E. Griswold and Richard L. Goerwitz.
procedure lcomb(L,i)			#: list combinations
local j
 
if i < 1 then fail
suspend if i = 1 then [!L]
else [L[j := 1 to *L - i + 1]] ||| lcomb(L[j + 1:0],i - 1)
 
end
Sample output:
3 combinations of 5 integers starting from 0
Intial list
[ 0 1 2 3 4 ]
Combinations:
[ 0 1 2 ]
[ 0 1 3 ]
[ 0 1 4 ]
[ 0 2 3 ]
[ 0 2 4 ]
[ 0 3 4 ]
[ 1 2 3 ]
[ 1 2 4 ]
[ 1 3 4 ]
[ 2 3 4 ]

[edit] J

[edit] Iteration

comb1=: dyad define
c=. 1 {.~ - d=. 1+y-x
z=. i.1 0
for_j. (d-1+y)+/&i.d do. z=. (c#j) ,. z{~;(-c){.&.><i.{.c=. +/\.c end.
)

[edit] Recursion

comb=: dyad define M.
if. (x>:y)+.0=x do. i.(x<:y),x else. (0,.x comb&.<: y),1+x comb y-1 end.
)

The M. uses memoization which greatly reduces the running time.

[edit] Brute Force

We can also generate all permutations and exclude those which are not properly sorted combinations. This is inefficient, but efficiency is not always important.

combb=: (#~ ((-:/:~)>/:~-:\:~)"1)@(# #: [: i. ^~)

[edit] Java

Translation of: JavaScript
Works with: Java version 1.5+
import java.util.Collections;
import java.util.LinkedList;
 
public class Comb{
 
public static void main(String[] args){
System.out.println(comb(3,5));
}
 
public static String bitprint(int u){
String s= "";
for(int n= 0;u > 0;++n, u>>= 1)
if((u & 1) > 0) s+= n + " ";
return s;
}
 
public static int bitcount(int u){
int n;
for(n= 0;u > 0;++n, u&= (u - 1));//Turn the last set bit to a 0
return n;
}
 
public static LinkedList<String> comb(int c, int n){
LinkedList<String> s= new LinkedList<String>();
for(int u= 0;u < 1 << n;u++)
if(bitcount(u) == c) s.push(bitprint(u));
Collections.sort(s);
return s;
}
}

[edit] JavaScript

function bitprint(u) {
var s="";
for (var n=0; u; ++n, u>>=1)
if (u&1) s+=n+" ";
return s;
}
function bitcount(u) {
for (var n=0; u; ++n, u=u&(u-1));
return n;
}
function comb(c,n) {
var s=[];
for (var u=0; u<1<<n; u++)
if (bitcount(u)==c)
s.push(bitprint(u))
return s.sort();
}
comb(3,5)

Alternative recursive version using and an array of values instead of length:

Translation of: Python
function combinations(arr, k){
var i,
subI,
ret = [],
sub,
next;
for(i = 0; i < arr.length; i++){
if(k === 1){
ret.push( [ arr[i] ] );
}else{
sub = combinations(arr.slice(i+1, arr.length), k-1);
for(subI = 0; subI < sub.length; subI++ ){
next = sub[subI];
next.unshift(arr[i]);
ret.push( next );
}
}
}
return ret;
}
combinations([0,1,2,3,4], 3);
// produces: [[0, 1, 2], [0, 1, 3], [0, 1, 4], [0, 2, 3], [0, 2, 4], [0, 3, 4], [1, 2, 3], [1, 2, 4], [1, 3, 4], [2, 3, 4]]
 
combinations(["Crosby", "Stills", "Nash", "Young"], 3);
// produces: [["Crosby", "Stills", "Nash"], ["Crosby", "Stills", "Young"], ["Crosby", "Nash", "Young"], ["Stills", "Nash", "Young"]]
 

[edit] jq

combination(r) generates a stream of combinations of the input array. The stream can be captured in an array as shown in the second example.

def combination(r):
if r > length or r < 0 then empty
elif r == length then .
else ( [.[0]] + (.[1:]|combination(r-1))),
( .[1:]|combination(r))
end;
 
# select r integers from the set (0 .. n-1)
def combinations(n;r): [range(0;n)] | combination(r);

Example 1

combinations(5;3) 
Output:
[0,1,2]
[0,1,3]
[0,1,4]
[0,2,3]
[0,2,4]
[0,3,4]
[1,2,3]
[1,2,4]
[1,3,4]
[2,3,4]

Example 2

["a", "b", "c", "d", "e"] | combination(3) ] | length
Output:
10

[edit] Julia

There is a built-in function combinations that generates an iterable sequence of the combinations that you can loop over. (Note that the combinations are computed on the fly during the loop iteration, and are not pre-computed or stored since there many be a very large number of them.)

for i in combinations(1:5,3)
print(i')
end
Output:
1	2	3
1	2	4
1	2	5
1	3	4
1	3	5
1	4	5
2	3	4
2	3	5
2	4	5
3	4	5

[edit]

to comb :n :list
if :n = 0 [output [[]]]
if empty? :list [output []]
output sentence map [sentence first :list ?] comb :n-1 bf :list ~
comb :n bf :list
end
print comb 3 [0 1 2 3 4]

[edit] Lua

 
-- Recursive version
function map(f, a, ...) if a then return f(a), map(f, ...) end end
function incr(k) return function(a) return k > a and a or a+1 end end
function combs(m, n)
if m * n == 0 then return {{}} end
local ret, old = {}, combs(m-1, n-1)
for i = 1, n do
for k, v in ipairs(old) do ret[#ret+1] = {i, map(incr(i), unpack(v))} end
end
return ret
end
 
for k, v in ipairs(combs(3, 5)) do print(unpack(v)) end
 
-- Iterative version
function icombs(a,b)
if a==0 then return end
local taken = {} local slots = {}
for i=1,a do slots[i]=0 end
for i=1,b do taken[i]=false end
local index = 1
while index > 0 do repeat
repeat slots[index] = slots[index] + 1
until slots[index] > b or not taken[slots[index]]
if slots[index] > b then
slots[index] = 0
index = index - 1
if index > 0 then
taken[slots[index]] = false
end
break
else
taken[slots[index]] = true
end
if index == a then
for i=1,a do io.write(slots[i]) io.write(" ") end
io.write("\n")
taken[slots[index]] = false
break
end
index = index + 1
until true end
end
 
icombs(3, 5)
 

[edit] Maple

This is built-in in Maple:

> combinat:-choose( 5, 3 );
[[1, 2, 3], [1, 2, 4], [1, 2, 5], [1, 3, 4], [1, 3, 5], [1, 4, 5], [2, 3, 4], [2, 3, 5],
 
[2, 4, 5], [3, 4, 5]]
 

[edit] Mathematica

combinations[n_Integer, m_Integer]/;m>= 0:=Union[Sort /@ Permutations[Range[0, n - 1], {m}]]

[edit] M4

divert(-1)
define(`set',`define(`$1[$2]',`$3')')
define(`get',`defn(`$1[$2]')')
define(`setrange',`ifelse(`$3',`',$2,`define($1[$2],$3)`'setrange($1,
incr($2),shift(shift(shift($@))))')')
define(`for',
`ifelse($#,0,``$0'',
`ifelse(eval($2<=$3),1,
`pushdef(`$1',$2)$4`'popdef(`$1')$0(`$1',incr($2),$3,`$4')')')')
define(`show',
`for(`k',0,decr($1),`get(a,k) ')')
 
define(`chklim',
`ifelse(get(`a',$3),eval($2-($1-$3)),
`chklim($1,$2,decr($3))',
`set(`a',$3,incr(get(`a',$3)))`'for(`k',incr($3),decr($2),
`set(`a',k,incr(get(`a',decr(k))))')`'nextcomb($1,$2)')')
define(`nextcomb',
`show($1)
ifelse(eval(get(`a',0)<$2-$1),1,
`chklim($1,$2,decr($1))')')
define(`comb',
`for(`j',0,decr($1),`set(`a',j,j)')`'nextcomb($1,$2)')
divert
 
comb(3,5)

[edit] MATLAB

This a built-in function in MATLAB called "nchoosek(n,k)". The argument "n" is a vector of values from which the combinations are made, and "k" is a scalar representing the amount of values to include in each combination.

Task Solution:

>> nchoosek((0:4),3)
 
ans =
 
0 1 2
0 1 3
0 1 4
0 2 3
0 2 4
0 3 4
1 2 3
1 2 4
1 3 4
2 3 4

[edit] Maxima

next_comb(n, p, a) := block(
[a: copylist(a), i: p],
if a[1] + p = n + 1 then return(und),
while a[i] - i >= n - p do i: i - 1,
a[i]: a[i] + 1,
for j from i + 1 thru p do a[j]: a[j - 1] + 1,
a
)$
 
combinations(n, p) := block(
[a: makelist(i, i, 1, p), v: [ ]],
while a # 'und do (v: endcons(a, v), a: next_comb(n, p, a)),
v
)$
 
combinations(5, 3);
/* [[1, 2, 3],
[1, 2, 4],
[1, 2, 5],
[1, 3, 4],
[1, 3, 5],
[1, 4, 5],
[2, 3, 4],
[2, 3, 5],
[2, 4, 5],
[3, 4, 5]] */

[edit] Nimrod

iterator comb(m, n): seq[int] =
var c = newSeq[int](n)
for i in 0 .. <n: c[i] = i
 
block outer:
while true:
yield c
 
var i = n-1
inc c[i]
if c[i] <= m - 1: continue
 
while c[i] >= m - n + i:
dec i
if i < 0: break outer
inc c[i]
while i < n-1:
c[i+1] = c[i] + 1
inc i
 
for i in comb(5, 3): echo i

Output:

@[0, 1, 2]
@[0, 1, 3]
@[0, 1, 4]
@[0, 2, 3]
@[0, 2, 4]
@[0, 3, 4]
@[1, 2, 3]
@[1, 2, 4]
@[1, 3, 4]
@[2, 3, 4]

[edit] OCaml

Like the Haskell code:

let rec comb m lst =
match m, lst with
0, _ -> [[]]
| _, [] -> []
| m, x :: xs -> List.map (fun y -> x :: y) (comb (pred m) xs) @
comb m xs
;;
comb 3 [0;1;2;3;4];;

Dynamic Programming solution:

let comb m xs =
let xs = Array.of_list xs in
if m > Array.length xs then
[]
else begin
let arr = Array.make (m+1) [] in
arr.(0) <- [[]];
for j = 0 to Array.length xs - m do
for i = 1 to m do
arr.(i) <- arr.(i) @ List.map (fun ys -> xs.(j+i-1)::ys) arr.(i-1)
done
done;
arr.(m)
end
;;
comb 3 [0;1;2;3;4];;

[edit] Octave

nchoosek([0:4], 3)

[edit] Oz

This can be implemented as a trivial application of finite set constraints:

declare
fun {Comb M N}
proc {CombScript Comb}
%% Comb is a subset of [0..N-1]
Comb = {FS.var.upperBound {List.number 0 N-1 1}}
%% Comb has cardinality M
{FS.card Comb M}
%% enumerate all possibilities
{FS.distribute naive [Comb]}
end
in
%% Collect all solutions and convert to lists
{Map {SearchAll CombScript} FS.reflect.upperBoundList}
end
in
{Inspect {Comb 3 5}}

[edit] PARI/GP

c(n,k,r,d)={
if(d==k,
for(i=2,k+1,
print1(r[i]" "));
print
,
for(i=r[d+1]+1,n,
r[d+2]=i;
c(n,k,r,d+1)));
}
 
c(5,3,vector(5,i,i-1),0)
 

[edit] Pascal

Program Combinations;
 
const
m_max = 3;
n_max = 5;
var
combination: array [1..m_max] of integer;
 
procedure generate(m: integer);
var
n, i: integer;
begin
if (m > m_max) then
begin
for i := 1 to m_max do
write (combination[i], ' ');
writeln;
end
else
for n := 1 to n_max do
if ((m = 1) or (n > combination[m-1])) then
begin
combination[m] := n;
generate(m + 1);
end;
end;
 
begin
generate(1);
end.

output

1 2 3 
1 2 4 
1 2 5 
1 3 4 
1 3 5 
1 4 5 
2 3 4 
2 3 5 
2 4 5 
3 4 5 

[edit] Perl

The ntheory module has a combinations iterator that runs in lexicographic order.

Library: ntheory
use ntheory qw/forcomb/;
forcomb { print "@_\n" } 5,3
Output:
0 1 2
0 1 3
0 1 4
0 2 3
0 2 4
0 3 4
1 2 3
1 2 4
1 3 4
2 3 4

Algorithm::Combinatorics also does lexicographic order and can return the whole array or an iterator:

use Algorithm::Combinatorics qw/combinations/;
my @c = combinations( [0..4], 3 );
print "@$_\n" for @c;
use Algorithm::Combinatorics qw/combinations/;
my $iter = combinations([0..4],3);
while (my $c = $iter->next) {
print "@$c\n";
}

Math::Combinatorics is another option but results will not be in lexicographic order as specified by the task.

[edit] Perl5i

Use a recursive solution, derived from the Perl6 (Haskell) solution

  • If we run out of eligable characters, we've gone too far, and won't find a solution along this path.
  • If we are looking for a single character, each character in @set is elegible, so return each as the single element of an array.
  • We have not yet reached the last character, so there are two possibilities:
    1. push the first element of the set onto the front of an N-1 length combination from the remainder of the set.
    2. skip the current element, and generate an N-length combination from the remainder

The major Perl5i -isms are the implicit "autoboxing" of the intermediate resulting array into an array object, with the use of unshift() as a method, and the "func" keyword and signature. Note that Perl can construct ranges of numbers or of letters, so it is natural to identify the characters as 'a' .. 'e'.

 
use perl5i::2;
 
# ----------------------------------------
# generate combinations of length $n consisting of characters
# from the sorted set @set, using each character once in a
# combination, with sorted strings in sorted order.
#
# Returns a list of array references, each containing one combination.
#
func combine($n, @set) {
return unless @set;
return map { [ $_ ] } @set if $n == 1;
 
my ($head) = shift @set;
my @result = combine( $n-1, @set );
for my $subarray ( @result ) {
$subarray->unshift( $head );
}
return ( @result, combine( $n, @set ) );
}
 
say @$_ for combine( 3, ('a'..'e') );
 

Output

abc
abd
abe
acd
ace
ade
bcd
bce
bde
cde

[edit] Perl 6

Works with: rakudo version 2014-01-24

There actually is a builtin:

.say for combinations(5,3);
Output:
0 1 2
0 1 3
0 1 4
0 2 3
0 2 4
0 3 4
1 2 3
1 2 4
1 3 4
2 3 4

Here is an iterative routine with the same output:

sub combinations(Int $n, Int $k) {
return ([],) unless $k;
return if $k > $n || $n <= 0;
my @c = ^$k;
gather loop {
take [@c];
next if @c[$k-1]++ < $n-1;
my $i = $k-2;
$i-- while $i >= 0 && @c[$i] >= $n-($k-$i);
last if $i < 0;
@c[$i]++;
while ++$i < $k { @c[$i] = @c[$i-1] + 1; }
}
}
.say for combinations(5,3);

or recursive with same output, but much slower:

sub combinations(Int $n, Int $k) {
return [] if $k == 0;
return () if $k > $n;
gather {
take [0, (1..^$n)[@$_]] for combinations($n-1, $k-1);
take [(1..^$n)[@$_]] for combinations($n-1, $k );
}
}
 
.say for combinations(5, 3);

[edit] PicoLisp

Translation of: Scheme
(de comb (M Lst)
(cond
((=0 M) '(NIL))
((not Lst))
(T
(conc
(mapcar
'((Y) (cons (car Lst) Y))
(comb (dec M) (cdr Lst)) )
(comb M (cdr Lst)) ) ) ) )
 
(comb 3 (1 2 3 4 5))

[edit] Pop11

Natural recursive solution: first we choose first number i and then we recursively generate all combinations of m - 1 numbers between i + 1 and n - 1. Main work is done in the internal 'do_combs' function, the outer 'comb' just sets up variable to accumulate results and reverses the final result.

The 'el_lst' parameter to 'do_combs' contains partial combination (list of numbers which were chosen in previous steps) in reverse order.

define comb(n, m);
lvars ress = [];
define do_combs(l, m, el_lst);
lvars i;
if m = 0 then
cons(rev(el_lst), ress) -> ress;
else
for i from l to n - m do
do_combs(i + 1, m - 1, cons(i, el_lst));
endfor;
endif;
enddefine;
do_combs(0, m, []);
rev(ress);
enddefine;
 
comb(5, 3) ==>

[edit] Prolog

The solutions work with SWI-Prolog
Solution with library clpfd : we first create a list of M elements, we say that the members of the list are numbers between 1 and N and there are in ascending order, finally we ask for a solution.

:- use_module(library(clpfd)).
 
comb_clpfd(L, M, N) :-
length(L, M),
L ins 1..N,
chain(L, #<),
label(L).

output :

 ?- comb_clpfd(L, 3, 5), writeln(L), fail.
[1,2,3]
[1,2,4]
[1,2,5]
[1,3,4]
[1,3,5]
[1,4,5]
[2,3,4]
[2,3,5]
[2,4,5]
[3,4,5]
false.

Another solution :

comb_Prolog(L, M, N) :-
length(L, M),
fill(L, 1, N).
 
fill([], _, _).
 
fill([H | T], Min, Max) :-
between(Min, Max, H),
H1 is H + 1,
fill(T, H1, Max).
 

with the same output.

[edit] List comprehension

Works with SWI-Prolog, library clpfd from Markus Triska, and list comprehension (see List comprehensions ).

:- use_module(library(clpfd)).
comb_lstcomp(N, M, V) :-
V <- {L & length(L, N), L ins 1..M & all_distinct(L), chain(L, #<), label(L)}.
 

Output :

2?- comb_lstcomp(3, 5, V).
V = [[1,2,3],[1,2,4],[1,2,5],[1,3,4],[1,3,5],[1,4,5],[2,3,4],[2,3,5],[2,4,5],[3,4,5]] ;
false.

[edit] Pure

comb m n = comb m (0..n-1) with
comb 0 _ = [[]];
comb _ [] = [];
comb m (x:xs) = [x:xs | xs = comb (m-1) xs] + comb m xs;
end;
 
comb 3 5;

[edit] PureBasic

Procedure.s Combinations(amount, choose)
NewList comb.s()
; all possible combinations with {amount} Bits
For a = 0 To 1 << amount
count = 0
; count set bits
For x = 0 To amount
If (1 << x)&a
count + 1
EndIf
Next
; if set bits are equal to combination length
; we generate a String representing our combination and add it to list
If count = choose
string$ = ""
For x = 0 To amount
If (a >> x)&1
; replace x by x+1 to start counting with 1
String$ + Str(x) + " "
EndIf
Next
AddElement(comb())
comb() = string$
EndIf
Next
; now we sort our list and format it for output as string
SortList(comb(), #PB_Sort_Ascending)
ForEach comb()
out$ + ", [ " + comb() + "]"
Next
ProcedureReturn Mid(out$, 3)
EndProcedure
 
Debug Combinations(5, 3)

[edit] Python

Starting from Python 2.6 and 3.0 you have a pre-defined function that returns an iterator. Here we turn the result into a list for easy printing:

>>> from itertools import combinations
>>> list(combinations(range(5),3))
[(0, 1, 2), (0, 1, 3), (0, 1, 4), (0, 2, 3), (0, 2, 4), (0, 3, 4), (1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4)]

Earlier versions could use functions like the following:

Translation of: E
def comb(m, lst):
if m == 0:
return [[]]
else:
return [[x] + suffix for i, x in enumerate(lst)
for suffix in comb(m - 1, lst[i + 1:])]

Example:

>>> comb(3, range(5))
[[0, 1, 2], [0, 1, 3], [0, 1, 4], [0, 2, 3], [0, 2, 4], [0, 3, 4], [1, 2, 3], [1, 2, 4], [1, 3, 4], [2, 3, 4]]
Translation of: Haskell
def comb(m, s):
if m == 0: return [[]]
if s == []: return []
return [s[:1] + a for a in comb(m-1, s[1:])] + comb(m, s[1:])
 
print comb(3, range(5))

[edit] Racket

Translation of: Haskell
 
(define sublists
(match-lambda**
[(0 _) '(())]
[(_ '()) '()]
[(m (cons x xs)) (append (map (curry cons x) (sublists (- m 1) xs))
(sublists m xs))]))
 
(define (combinations n m)
(sublists n (range m)))
 

Output:

> (combinations 3 5)
'((0 1 2)
  (0 1 3)
  (0 1 4)
  (0 2 3)
  (0 2 4)
  (0 3 4)
  (1 2 3)
  (1 2 4)
  (1 3 4)
  (2 3 4))

[edit] R

print(combn(0:4, 3))

Combinations are organized per column, so to provide an output similar to the one in the task text, we need the following:

r <- combn(0:4, 3)
for(i in 1:choose(5,3)) print(r[,i])

[edit] REXX

This REXX program supports up to 62 symbols (one symbol for each "thing").
It supports any number of "things" beyond the 62 symbols by using the actual number instead of a symbol.

/*REXX program shows combination sets for  X  things taken  Y  at a time*/
@abc='abcdefghijklmnopqrstuvwxyz'; @abcU=@abc; upper @abcU; @digs=123456789
parse arg x y symbols .; if x=='' | x==',' then x=5
if y=='' | y==',' then y=3
if symbols=='' then symbols=@digs||@abc||@abcU /*symbol table string.*/
say "────────────" x 'things taken' y "at a time:"
say "────────────" combN(x,y) 'combinations.'
exit /*stick a fork in it, we're done.*/
/*──────────────────────────────────COMBN subroutine────────────────────*/
combN: procedure expose symbols; parse arg x,y; base=x+1; bbase=base-y
!.=0; do i=1 for y;  !.i=i
end /*i*/
 
do j=1; L=; do d=1 for y
L=L word(substr(symbols,!.d,1)  !.d,1)
end /*d*/
say L
 !.y=!.y+1; if !.y==base then if .combUp(y-1) then leave
end /*j*/
return j
 
.combUp: procedure expose !. y bbase; parse arg d; if d==0 then return 1
p=!.d; do u=d to y;  !.u=p+1
if !.u==bbase+u then return .combUp(u-1)
p=!.u
end /*u*/
return 0

output when the following was specified: 5 3 01234

──────────── 5 things taken 3 at a time:
 0 1 2
 0 1 3
 0 1 4
 0 2 3
 0 2 4
 0 3 4
 1 2 3
 1 2 4
 1 3 4
 2 3 4
──────────── 10 combinations.

output when the following was specified: 5 3 abcde

──────────── 5 things taken 3 at a time:
 a b c
 a b d
 a b e
 a c d
 a c e
 a d e
 b c d
 b c e
 b d e
 c d e
──────────── 10 combinations.

[edit] Ruby

Works with: Ruby version 1.8.7+
def comb(m, n)
(0...n).to_a.combination(m).to_a
end
 
comb(3, 5) # => [[0, 1, 2], [0, 1, 3], [0, 1, 4], [0, 2, 3], [0, 2, 4], [0, 3, 4], [1, 2, 3], [1, 2, 4], [1, 3, 4], [2, 3, 4]]

[edit] Rust

Works with: Rust version 0.9
 
fn comb<T: std::fmt::Default>(arr: &[T], n: uint) {
let mut incl_arr: ~[bool] = std::vec::from_elem(arr.len(), false);
comb_intern(arr, n, incl_arr, 0);
}
 
fn comb_intern<T: std::fmt::Default>(arr: &[T], n: uint, incl_arr: &mut [bool], index: uint) {
if (arr.len() < n + index) { return; }
if (n == 0) {
let mut it = arr.iter().zip(incl_arr.iter()).filter_map(|(val, incl)|
if (*incl) { Some(val) } else { None }
);
for val in it { print!("{} ", *val); }
print("\n");
return;
}
 
incl_arr[index] = true;
comb_intern(arr, n-1, incl_arr, index+1);
incl_arr[index] = false;
 
comb_intern(arr, n, incl_arr, index+1);
}
 
fn main() {
let arr1 = ~[1, 2, 3, 4, 5];
comb(arr1, 3);
 
let arr2 = ~["A", "B", "C", "D", "E"];
comb(arr2, 3);
}
 

[edit] Scala

implicit def toComb(m: Int) = new AnyRef {
def comb(n: Int) = recurse(m, List.range(0, n))
private def recurse(m: Int, l: List[Int]): List[List[Int]] = (m, l) match {
case (0, _) => List(Nil)
case (_, Nil) => Nil
case _ => (recurse(m - 1, l.tail) map (l.head :: _)) ::: recurse(m, l.tail)
}
}

Usage:

scala> 3 comb 5
res170: List[List[Int]] = List(List(0, 1, 2), List(0, 1, 3), List(0, 1, 4), List(0, 2, 3), List(0, 2, 4), List(0, 3, 4),
 List(1, 2, 3), List(1, 2, 4), List(1, 3, 4), List(2, 3, 4))

[edit] Scala 2.9.x

Works with: Scala version 2.9.1
scala> (0 to 4 toList) combinations(3) toList
res1: List[List[Int]] = List(List(0, 1, 2), List(0, 1, 3), List(0, 1, 4), List(0, 2, 3), List(0, 2, 4), List(0, 3, 4), List(1, 2, 3), List(1, 2, 4), List(1, 3, 4), List(2, 3, 4))
 

[edit] Scheme

Like the Haskell code:

(define (comb m lst)
(cond ((= m 0) '(()))
((null? lst) '())
(else (append (map (lambda (y) (cons (car lst) y))
(comb (- m 1) (cdr lst)))
(comb m (cdr lst))))))
 
(comb 3 '(0 1 2 3 4))

[edit] Seed7

$ include "seed7_05.s7i";
 
const type: combinations is array array integer;
 
const func combinations: comb (in array integer: arr, in integer: k) is func
result
var combinations: combResult is combinations.value;
local
var integer: x is 0;
var integer: i is 0;
var array integer: suffix is 0 times 0;
begin
if k = 0 then
combResult := 1 times 0 times 0;
else
for x key i range arr do
for suffix range comb(arr[succ(i) ..], pred(k)) do
combResult &:= [] (x) & suffix;
end for;
end for;
end if;
end func;
 
const proc: main is func
local
var array integer: aCombination is 0 times 0;
var integer: element is 0;
begin
for aCombination range comb([] (0, 1, 2, 3, 4), 3) do
for element range aCombination do
write(element lpad 3);
end for;
writeln;
end for;
end func;

Output:

  0  1  2
  0  1  3
  0  1  4
  0  2  3
  0  2  4
  0  3  4
  1  2  3
  1  2  4
  1  3  4
  2  3  4

[edit] SETL

print({0..4} npow 3);

[edit] Smalltalk

Works with: Pharo version 1.3-13315
 
(0 to: 4)
combinations: 3 atATimeDo: [ :x |
':-)' logCr: x ].
 
"output on Transcript:
#(0 1 2)
#(0 1 3)
#(0 1 4)
#(0 2 3)
#(0 2 4)
#(0 3 4)
#(1 2 3)
#(1 2 4)
#(1 3 4)
#(2 3 4)"

 

[edit] Standard ML

fun comb (0, _    ) = [[]]
| comb (_, [] ) = []
| comb (m, x::xs) = map (fn y => x :: y) (comb (m-1, xs)) @
comb (m, xs)
;
comb (3, [0,1,2,3,4]);

[edit] Tcl

ref[1]

proc comb {m n} {
set set [list]
for {set i 0} {$i < $n} {incr i} {lappend set $i}
return [combinations $set $m]
}
proc combinations {list size} {
if {$size == 0} {
return [list [list]]
}
set retval {}
for {set i 0} {($i + $size) <= [llength $list]} {incr i} {
set firstElement [lindex $list $i]
set remainingElements [lrange $list [expr {$i + 1}] end]
foreach subset [combinations $remainingElements [expr {$size - 1}]] {
lappend retval [linsert $subset 0 $firstElement]
}
}
return $retval
}
 
comb 3 5 ;# ==> {0 1 2} {0 1 3} {0 1 4} {0 2 3} {0 2 4} {0 3 4} {1 2 3} {1 2 4} {1 3 4} {2 3 4}

[edit] TXR

TXR has repeating and non-repeating permutation and combination functions that produce lazy lists. They are generic over lists, strings and vectors. In addition, the combinations function also works over hashes.

Combinations and permutations are produced in lexicographic order (except in the case of hashes).

@(do 
(defun comb-n-m (n m)
(comb (range* 0 n) m))
 
(put-line `3 comb 5 = @(comb-n-m 5 3)`))

Run:

$ txr combinations.txr 
3 comb 5 = ((0 1 2) (0 1 3) (0 1 4) (0 2 3) (0 2 4) (0 3 4) (1 2 4) (1 3 4) (2 3 4))

[edit] Ursala

Most of the work is done by the standard library function choices, whose implementation is shown here for the sake of comparison with other solutions,

choices = ^(iota@r,~&l); leql@a^& ~&al?\&! ~&arh2fabt2RDfalrtPXPRT

where leql is the predicate that compares list lengths. The main body of the algorithm (~&arh2fabt2RDfalrtPXPRT) concatenates the results of two recursive calls, one of which finds all combinations of the required size from the tail of the list, and the other of which finds all combinations of one less size from the tail, and then inserts the head into each. choices generates combinations of an arbitrary set but not necessarily in sorted order, which can be done like this.

#import std
#import nat
 
combinations = @rlX choices^|(iota,~&); -< @p nleq+ ==-~rh
  • The sort combinator (-<) takes a binary predicate to a function that sorts a list in order of that predicate.
  • The predicate in this case begins by zipping its two arguments together with @p.
  • The prefiltering operator -~ scans a list from the beginning until it finds the first item to falsify a predicate (in this case equality, ==) and returns a pair of lists with the scanned items satisfying the predicate on the left and the remaining items on the right.
  • The rh suffix on the -~ operator causes it to return only the head of the right list as its result, which in this case will be the first pair of unequal items in the list.
  • The nleq function then tests whether the left side of this pair is less than or equal to the right.
  • The overall effect of using everything starting from the @p as the predicate to a sort combinator is therefore to sort a list of lists of natural numbers according to the order of the numbers in the first position where they differ.

test program:

#cast %nLL
 
example = combinations(3,5)

output:

<
   <0,1,2>,
   <0,1,3>,
   <0,1,4>,
   <0,2,3>,
   <0,2,4>,
   <0,3,4>,
   <1,2,3>,
   <1,2,4>,
   <1,3,4>,
   <2,3,4>>

[edit] V

like scheme (using variables)

[comb [m lst] let
[ [m zero?] [[[]]]
[lst null?] [[]]
[true] [m pred lst rest comb [lst first swap cons] map
m lst rest comb concat]
] when].

Using destructuring view and stack not *pure at all

[comb
[ [pop zero?] [pop pop [[]]]
[null?] [pop pop []]
[true] [ [m lst : [m pred lst rest comb [lst first swap cons] map
m lst rest comb concat]] view i ]
] when].

Pure concatenative version

[comb
[2dup [a b : a b a b] view].
[2pop pop pop].
 
[ [pop zero?] [2pop [[]]]
[null?] [2pop []]
[true] [2dup [pred] dip uncons swapd comb [cons] map popd rollup rest comb concat]
] when].

Using it

|3 [0 1 2 3 4] comb
=[[0 1 2] [0 1 3] [0 1 4] [0 2 3] [0 2 4] [0 3 4] [1 2 3] [1 2 4] [1 3 4] [2 3 4]]

[edit] XPL0

code ChOut=8, CrLf=9, IntOut=11;
def M=3, N=5;
int A(N-1);
 
proc Combos(D, S); \Display all size M combinations of N in sorted order
int D, S; \depth of recursion, starting value of N
int I;
[if D<M then \depth < size
for I:= S to N-1 do
[A(D):= I;
Combos(D+1, I+1);
]
else [for I:= 0 to M-1 do
[IntOut(0, A(I)); ChOut(0, ^ )];
CrLf(0);
];
];
 
Combos(0, 0)

Output:

0 1 2 
0 1 3 
0 1 4 
0 2 3 
0 2 4 
0 3 4 
1 2 3 
1 2 4 
1 3 4 
2 3 4 

[edit] zkl

Translation of: OCaml
fcn comb(k,seq){	// no repeats, seq is finite
seq=seq.makeReadOnly(); // because I append to parts of seq
fcn(k,seq){
if(k<=0) return(T(T));
if(not seq) return(T);
self.fcn(k-1,seq[1,*]).pump(List,seq[0,1].extend)
.extend(self.fcn(k,seq[1,*]));
}(k,seq);
}
comb(3,"abcde".split("")).apply("concat")
Output:
L("abc","abd","abe","acd","ace","ade","bcd","bce","bde","cde")
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