⚠️ Warning: This is a draft ⚠️

This means it might contain formatting issues, incorrect code, conceptual problems, or other severe issues.

If you want to help to improve and eventually enable this page, please fork RosettaGit's repository and open a merge request on GitHub.

{{draft task|Probability and statistics}}

;Task: '''Generate''' a sequence of permutations of n elements drawn from choice of k values.

This sequence will have $n^k$ elements, unless the program decides to terminate early.

Do not store all the intermediate values of the sequence, rather generate them as required, and pass the intermediate result to a deciding routine for combinations selection and/or early generator termination.

For example: When "cracking" a ''"combination"'' lock a sequence is required, but the sequence is terminated once a successful ''"combination"'' is found. This case is a good example of where it is not required to store all the intermediate '''permutations'''.

## AppleScript

### Strict evaluation of the whole set

Permutations with repetitions, using strict evaluation, generating the entire set (where system constraints permit) with some degree of efficiency. For lazy or interruptible evaluation, see the second example below.

```-- e.g. replicateM(3, {1, 2})) ->
-- {{1, 1, 1}, {1, 1, 2}, {1, 2, 1}, {1, 2, 2}, {2, 1, 1},
--  {2, 1, 2}, {2, 2, 1}, {2, 2, 2}}

-- replicateM :: Int -> [a] -> [[a]]
on replicateM(n, xs)
script go
script cons
on |λ|(a, bs)
{a} & bs
end |λ|
end script
on |λ|(x)
if x ≤ 0 then
{{}}
else
liftA2List(cons, xs, |λ|(x - 1))
end if
end |λ|
end script

go's |λ|(n)
end replicateM

-- TEST ------------------------------------------------------------
on run

replicateM(2, {1, 2, 3})

-- {{1, 1}, {1, 2}, {1, 3}, {2, 1}, {2, 2}, {2, 3}, {3, 1}, {3, 2}, {3, 3}}
end run

-- GENERIC FUNCTIONS -----------------------------------------------

-- concatMap :: (a -> [b]) -> [a] -> [b]
on concatMap(f, xs)
set lng to length of xs
set acc to {}
tell mReturn(f)
repeat with i from 1 to lng
set acc to acc & |λ|(item i of xs, i, xs)
end repeat
end tell
return acc
end concatMap

-- liftA2List :: (a -> b -> c) -> [a] -> [b] -> [c]
on liftA2List(f, xs, ys)
script
property g : mReturn(f)'s |λ|
on |λ|(x)
script
on |λ|(y)
{g(x, y)}
end |λ|
end script
concatMap(result, ys)
end |λ|
end script
concatMap(result, xs)
end liftA2List

-- Lift 2nd class handler function into 1st class script wrapper
-- mReturn :: First-class m => (a -> b) -> m (a -> b)
on mReturn(f)
if class of f is script then
f
else
script
property |λ| : f
end script
end if
end mReturn
```

{{Out}}

```{{1, 1}, {1, 2}, {1, 3}, {2, 1}, {2, 2}, {2, 3}, {3, 1}, {3, 2}, {3, 3}}
```

### Lazy evaluation with a generator

Permutations with repetition by treating the $n^k$ elements as an ordered set, and writing a function from a zero-based index to the nth permutation. This allows us terminate a repeated generation on some condition, or explore a sub-set without needing to generate the whole set:

```use AppleScript version "2.4"
use framework "Foundation"

-- permutesWithRepns :: [a] -> Int -> Generator [[a]]
on permutesWithRepns(xs, n)
script
property f : curry3(my nthPermutationWithRepn)'s |λ|(xs)'s |λ|(n)
property limit : (length of xs) ^ n
property i : -1
on |λ|()
set i to 1 + i
if i < limit then
return f's |λ|(i)
else
missing value
end if
end |λ|
end script
end permutesWithRepns

-- nthPermutationWithRepn :: [a] -> Int -> Int -> [a]
on nthPermutationWithRepn(xs, intGroup, intIndex)
set intBase to length of xs
if intIndex < (intBase ^ intGroup) then
set ds to baseDigits(intBase, xs, intIndex)

-- With any 'leading zeros' required by length
replicate(intGroup - (length of ds), item 1 of xs) & ds
else
missing value
end if
end nthPermutationWithRepn

-- baseDigits :: Int -> [a] -> [a]
on baseDigits(intBase, digits, n)
script
on |λ|(v)
if 0 = v then
Nothing()
else
Just(Tuple(item (1 + (v mod intBase)) of digits, ¬
v div intBase))
end if
end |λ|
end script
unfoldr(result, n)
end baseDigits

-- TEST ------------------------------------------------------------------
on run
set cs to "ACKR"
set wordLength to 5
set gen to permutesWithRepns(cs, wordLength)

set i to 0
set v to gen's |λ|() -- First permutation drawn from series
set alpha to v
set psi to alpha

repeat while missing value is not v
set s to concat(v)
if "crack" = toLower(s) then
return ("Permutation " & (i as text) & " of " & ¬
(((length of cs) ^ wordLength) as integer) as text) & ¬
": " & s & linefeed & ¬
"Found after searching from " & alpha & " thru " & psi
else
set i to 1 + i
set psi to v
end if
set v to gen's |λ|()
end repeat
end run

-- GENERIC ----------------------------------------------------------

-- Just :: a -> Maybe a
on Just(x)
{type:"Maybe", Nothing:false, Just:x}
end Just

-- Nothing :: Maybe a
on Nothing()
{type:"Maybe", Nothing:true}
end Nothing

-- Tuple (,) :: a -> b -> (a, b)
on Tuple(a, b)
{type:"Tuple", |1|:a, |2|:b, length:2}
end Tuple

-- concat :: [[a]] -> [a]
-- concat :: [String] -> String
on concat(xs)
set lng to length of xs
if 0 < lng and string is class of (item 1 of xs) then
set acc to ""
else
set acc to {}
end if
repeat with i from 1 to lng
set acc to acc & item i of xs
end repeat
acc
end concat

-- curry3 :: ((a, b, c) -> d) -> a -> b -> c -> d
on curry3(f)
script
on |λ|(a)
script
on |λ|(b)
script
on |λ|(c)
|λ|(a, b, c) of mReturn(f)
end |λ|
end script
end |λ|
end script
end |λ|
end script
end curry3

-- Lift 2nd class handler function into 1st class script wrapper
-- mReturn :: First-class m => (a -> b) -> m (a -> b)
on mReturn(f)
if class of f is script then
f
else
script
property |λ| : f
end script
end if
end mReturn

-- Egyptian multiplication - progressively doubling a list, appending
-- stages of doubling to an accumulator where needed for binary
-- assembly of a target length
-- replicate :: Int -> a -> [a]
on replicate(n, a)
set out to {}
if n < 1 then return out
set dbl to {a}

repeat while (n > 1)
if (n mod 2) > 0 then set out to out & dbl
set n to (n div 2)
set dbl to (dbl & dbl)
end repeat
return out & dbl
end replicate

-- toLower :: String -> String
on toLower(str)
set ca to current application
((ca's NSString's stringWithString:(str))'s ¬
lowercaseStringWithLocale:(ca's NSLocale's currentLocale())) as text
end toLower

-- > unfoldr (\b -> if b == 0 then Nothing else Just (b, b-1)) 10
-- > [10,9,8,7,6,5,4,3,2,1]
-- unfoldr :: (b -> Maybe (a, b)) -> b -> [a]
on unfoldr(f, v)
set xr to Tuple(v, v) -- (value, remainder)
set xs to {}
tell mReturn(f)
repeat -- Function applied to remainder.
set mb to |λ|(|2| of xr)
if Nothing of mb then
exit repeat
else -- New (value, remainder) tuple,
set xr to Just of mb
-- and value appended to output list.
set end of xs to |1| of xr
end if
end repeat
end tell
return xs
end unfoldr
```

{{Out}}

```Permutation 589 of 1024: CRACK
Found after searching from AAAAA thru ARACK
```

## ALGOL 68

{{works with|ALGOL 68|Revision 1 - one minor extension to language used - PRAGMA READ, similar to C's #include directive.}} {{works with|ALGOL 68G|Any - tested with release [http://sourceforge.net/projects/algol68/files/algol68g/algol68g-2.6 algol68g-2.6].}} {{wont work with|ELLA ALGOL 68|Any (with appropriate job cards) - tested with release [http://sourceforge.net/projects/algol68/files/algol68toc/algol68toc-1.8.8d/algol68toc-1.8-8d.fc9.i386.rpm/download 1.8-8d] - due to extensive use of '''format'''[ted] ''transput''.}} '''File: prelude_permutations_with_repetitions.a68'''

```# -*- coding: utf-8 -*- #

MODE PERMELEMLIST = FLEX[0]PERMELEM;
MODE PERMELEMLISTYIELD = PROC(PERMELEMLIST)VOID;

PROC perm gen elemlist = (FLEX[]PERMELEMLIST master, PERMELEMLISTYIELD yield)VOID:(
[LWB master:UPB master]INT counter;
[LWB master:UPB master]PERMELEM out;
FOR i FROM LWB counter TO UPB counter DO
INT c = counter[i] := LWB master[i];
out[i] := master[i][c]
OD;
yield(out);
WHILE TRUE DO
INT next i := LWB counter;
counter[next i] +:= 1;
FOR i FROM LWB counter TO UPB counter WHILE counter[i]>UPB master[i] DO
INT c = counter[i] := LWB master[i];
out[i] := master[i][c];
next i := i + 1;
IF next i > UPB counter THEN done FI;
counter[next i] +:= 1
OD;
INT c = counter[next i];
out[next i] := master[next i][c];
yield(out)
OD;
done: SKIP
);

SKIP
```

'''File: test_permutations_with_repetitions.a68'''

```#!/usr/bin/a68g --script #
# -*- coding: utf-8 -*- #

MODE PERMELEM = STRING;
PR READ "prelude_permutations_with_repetitions.a68" PR;

INT lead actor = 1, co star = 2;
PERMELEMLIST actors list = ("Chris Ciaffa", "Keith Urban","Tom Cruise",
"Katie Holmes","Mimi Rogers","Nicole Kidman");

FLEX[0]PERMELEMLIST combination := (actors list, actors list, actors list, actors list);

FORMAT partner fmt = \$g"; "\$;
test:(
# FOR PERMELEMELEM candidate in # perm gen elemlist(combination #) DO (#,
##   (PERMELEMLIST candidate)VOID: (
printf((partner fmt,candidate));
IF candidate[lead actor] = "Keith Urban" AND candidate[co star]="Nicole Kidman" OR
candidate[co star] = "Keith Urban" AND candidate[lead actor]="Nicole Kidman" THEN
print((" => Sunday + Faith as extras", new line)); # children #
done
FI;
print(new line)
# OD #));
done: SKIP
)
```

'''Output:'''

```
Chris Ciaffa; Chris Ciaffa; Chris Ciaffa; Chris Ciaffa;
Keith Urban; Chris Ciaffa; Chris Ciaffa; Chris Ciaffa;
Tom Cruise; Chris Ciaffa; Chris Ciaffa; Chris Ciaffa;
Katie Holmes; Chris Ciaffa; Chris Ciaffa; Chris Ciaffa;
Mimi Rogers; Chris Ciaffa; Chris Ciaffa; Chris Ciaffa;
Nicole Kidman; Chris Ciaffa; Chris Ciaffa; Chris Ciaffa;
Chris Ciaffa; Keith Urban; Chris Ciaffa; Chris Ciaffa;
Keith Urban; Keith Urban; Chris Ciaffa; Chris Ciaffa;
Tom Cruise; Keith Urban; Chris Ciaffa; Chris Ciaffa;
Katie Holmes; Keith Urban; Chris Ciaffa; Chris Ciaffa;
Mimi Rogers; Keith Urban; Chris Ciaffa; Chris Ciaffa;
Nicole Kidman; Keith Urban; Chris Ciaffa; Chris Ciaffa;  => Sunday + Faith as extras

```

## AutoHotkey

Use the function from http://rosettacode.org/wiki/Permutations#Alternate_Version with opt=1

```P(n,k="",opt=0,delim="",str="") { ; generate all n choose k permutations lexicographically
;1..n = range, or delimited list, or string to parse
;	to process with a different min index, pass a delimited list, e.g. "0`n1`n2"
;k = length of result
;opt 0 = no repetitions
;opt 1 = with repetitions
;opt 2 = run for 1..k
;opt 3 = run for 1..k with repetitions
;str = string to prepend (used internally)
;returns delimited string, error message, or (if k > n) a blank string
i:=0
If !InStr(n,"`n")
If n in 2,3,4,5,6,7,8,9
Loop, %n%
n := A_Index = 1 ? A_Index : n "`n" A_Index
Else
Loop, Parse, n, %delim%
n := A_Index = 1 ? A_LoopField : n "`n" A_LoopField
If (k = "")
RegExReplace(n,"`n","",k), k++
If k is not Digit
Return "k must be a digit."
If opt not in 0,1,2,3
Return "opt invalid."
If k = 0
Return str
Else
Loop, Parse, n, `n
If (!InStr(str,A_LoopField) || opt & 1)
s .= (!i++ ? (opt & 2 ? str "`n" : "") : "`n" )
. P(n,k-1,opt,delim,str . A_LoopField . delim)
Return s
}
```

## C

```
#include <stdlib.h>

int main(){
int temp;
int numbers=3;
int a[numbers+1], upto = 4, temp2;
for( temp2 = 1 ; temp2 <= numbers; temp2++){
a[temp2]=1;
}
a[numbers]=0;
temp=numbers;
while(1){
if(a[temp]==upto){
temp--;
if(temp==0)
break;
}
else{
a[temp]++;
while(temp<numbers){
temp++;
a[temp]=1;
}

printf("(");
for( temp2 = 1 ; temp2 <= numbers; temp2++){
printf("%d", a[temp2]);
}
printf(")");
}
}
return 0;
}
```

{{out}}

```(111)(112)(113)(114)(121)(122)(123)(124)(131)(132)(133)(134)(141)(142)(143)(144)(211)(212)(213)(214)(221)(222)(223)(224)(231)(232)(233)(234)(241)(242)(243)(244)(311)(312)(313)(314)(321)(322)(323)(324)(331)(332)(333)(334)(341)(342)(343)(344)(411)(412)(413)(414)(421)(422)(423)(424)(431)(432)(433)(434)(441)(442)(443)(444)
```

## C++

```
#include <stdio.h>
#include <stdlib.h>

struct Generator
{
public:
Generator(int s, int v)
: cSlots(s)
, cValues(v)
{
a = new int[s];

for (int i = 0; i < cSlots - 1; i++) {
a[i] = 1;
}
a[cSlots - 1] = 0;

nextInd = cSlots;
}

~Generator()
{
delete a;
}

bool doNext()
{
for (;;)
{
if (a[nextInd - 1] == cValues) {
nextInd--;
if (nextInd == 0)
return false;
}
else {
a[nextInd - 1]++;
while (nextInd < cSlots) {
nextInd++;
a[nextInd - 1] = 1;
}

return true;
}
}
}

void doPrint()
{
printf("(");
for (int i = 0; i < cSlots; i++) {
printf("%d", a[i]);
}
printf(")");
}

private:
int *a;
int cSlots;
int cValues;
int nextInd;
};

int main()
{
Generator g(3, 4);

while (g.doNext()) {
g.doPrint();
}

return 0;
}

```

{{out}}

```
(111)(112)(113)(114)(121)(122)(123)(124)(131)(132)(133)(134)(141)(142)(143)(144)(211)(212)(213)(214)(221)(222)(223)(224)(231)(232)(233)(234)(241)(242)(243)(244)(311)(312)(313)(314)(321)(322)(323)(324)(331)(332)(333)(334)(341)(342)(343)(344)(411)(412)(413)(414)(421)(422)(423)(424)(431)(432)(433)(434)(441)(442)(443)(444)

```

## D

### opApply Version

{{trans|Scala}}

```import std.array;

struct PermutationsWithRepetitions(T) {
const T[] data;
const int n;

int opApply(int delegate(ref T[]) dg) {
int result;
T[] aux;

if (n == 1) {
foreach (el; data) {
aux = [el];
result = dg(aux);
if (result) goto END;
}
} else {
foreach (el; data) {
foreach (p; PermutationsWithRepetitions(data, n - 1)) {
aux = el ~ p;
result = dg(aux);
if (result) goto END;
}
}
}

END:
return result;
}
}

auto permutationsWithRepetitions(T)(T[] data, in int n) pure nothrow
in {
assert(!data.empty && n > 0);
} body {
return PermutationsWithRepetitions!T(data, n);
}

void main() {
import std.stdio, std.array;
[1, 2, 3].permutationsWithRepetitions(2).array.writeln;
}
```

{{out}}

```[[1, 1], [1, 2], [1, 3], [2, 1], [2, 2], [2, 3], [3, 1], [3, 2], [3, 3]]
```

### Generator Range Version

{{trans|Scala}}

```import std.stdio, std.array, std.concurrency;

Generator!(T[]) permutationsWithRepetitions(T)(T[] data, in uint n)
in {
assert(!data.empty && n > 0);
} body {
return new typeof(return)({
if (n == 1) {
foreach (el; data)
yield([el]);
} else {
foreach (el; data)
foreach (perm; permutationsWithRepetitions(data, n - 1))
yield(el ~ perm);
}
});
}

void main() {
[1, 2, 3].permutationsWithRepetitions(2).writeln;
}
```

The output is the same.

## EchoLisp

```
(lib 'sequences) ;; (indices   ..)
(lib 'list) ;; (list-permute ..)

;; (indices range_1 ..range_k) returns a  procrastinator (lazy sequence)
;; which gives all combinations of indices_i in range_i.
;;
;; If  all  k ranges are equal to (0 ...n-1)
;; (indices (make-vector k n))
;; will give the n^k permutations with repetitions of the integers (0 ... n-1).

(define perms (indices (make-vector 2 3)))
(take perms #:all)
→ (#(0 0) #(0 1) #(0 2) #(1 0) #(1 1) #(1 2) #(2 0) #(2 1) #(2 2))
(length perms) → 9

;; 6-permute the numbers (0 ....9)
(define perms (indices (make-vector 6 10)))
(length perms) → 1000000

;; passing the procrastinator to a routine
;; which stops when sum = 22
(for ((p perms))
#:break (= (apply + (vector->list p)) 22) => p )
→  #( 0 0 0 4 9 9)

;; to permute any objects, use (list-permute list permutation-vector/list)
(list-permute '(a b c d e) '(1 0 1 0 3 2 1))
→ (b a b a d c b)
(list-permute '(a b c d e) #(1 0 1 0 3 2 1))
→ (b a b a d c b)

```

## Elixir

{{trans|Erlang}}

```defmodule RC do
def perm_rep(list), do: perm_rep(list, length(list))

def perm_rep([], _), do: [[]]
def perm_rep(_,  0), do: [[]]
def perm_rep(list, i) do
for x <- list, y <- perm_rep(list, i-1), do: [x|y]
end
end

list = [1, 2, 3]
Enum.each(1..3, fn n ->
IO.inspect RC.perm_rep(list,n)
end)
```

{{out}}

```
[[1], [2], [3]]
[[1, 1], [1, 2], [1, 3], [2, 1], [2, 2], [2, 3], [3, 1], [3, 2], [3, 3]]
[[1, 1, 1], [1, 1, 2], [1, 1, 3], [1, 2, 1], [1, 2, 2], [1, 2, 3], [1, 3, 1],
[1, 3, 2], [1, 3, 3], [2, 1, 1], [2, 1, 2], [2, 1, 3], [2, 2, 1], [2, 2, 2],
[2, 2, 3], [2, 3, 1], [2, 3, 2], [2, 3, 3], [3, 1, 1], [3, 1, 2], [3, 1, 3],
[3, 2, 1], [3, 2, 2], [3, 2, 3], [3, 3, 1], [3, 3, 2], [3, 3, 3]]

```

## Erlang

```-module(permute).
-export([permute/1]).

permute(L) -> permute(L,length(L)).
permute([],_) -> [[]];
permute(_,0) -> [[]];
permute(L,I) -> [[X|Y] || X<-L, Y<-permute(L,I-1)].
```

## Go

```package main

import "fmt"

var (
n      = 3
values = []string{"A", "B", "C", "D"}
k      = len(values)
decide = func(p []string) bool {
return p[0] == "B" && p[1] == "C"
}
)

func main() {
pn := make([]int, n)
p := make([]string, n)
for {
// generate permutaton
for i, x := range pn {
p[i] = values[x]
}
// show progress
fmt.Println(p)
// pass to deciding function
if decide(p) {
return // terminate early
}
// increment permutation number
for i := 0; ; {
pn[i]++
if pn[i] < k {
break
}
pn[i] = 0
i++
if i == n {
return // all permutations generated
}
}
}
}
```

{{out}}

```
[A A A]
[B A A]
[C A A]
[D A A]
[A B A]
[B B A]
[C B A]
[D B A]
[A C A]
[B C A]

```

```import Control.Monad (replicateM)

main = mapM_ print (replicateM 2 [1,2,3])
```

{{out}}

```
[1,1]
[1,2]
[1,3]
[2,1]
[2,2]
[2,3]
[3,1]
[3,2]
[3,3]

```

## J

Position in the sequence is an integer from `i.n^k`, for example:

```   i.3^2
0 1 2 3 4 5 6 7 8
```

The sequence itself is expressed using `(k#n)#: position`, for example:

```   (2#3)#:i.3^2
0 0
0 1
0 2
1 0
1 1
1 2
2 0
2 1
2 2
```

Partial sequences belong in a context where they are relevant and the sheer number of such possibilities make it inadvisable to generalize outside of those contexts. But anything that can generate integers will do. For example:

```   (2#3)#:3 4 5
1 0
1 1
1 2
```

We might express this as a verb

```perm=: # #: i.@^~
```

with example use:

```   2 perm 3
0 0
0 1
0 2
1 0
...
```

but the structural requirements of this task (passing intermediate results "when needed") mean that we are not looking for a word that does it all, but are instead looking for components that we can assemble in other contexts. This means that the language primitives are what's needed here.

## Java

{{works with|Java|8}}

```import java.util.function.Predicate;

public class PermutationsWithRepetitions {

public static void main(String[] args) {
char[] chars = {'a', 'b', 'c', 'd'};
// looking for bba
permute(chars, 3, i -> i[0] == 1 && i[1] == 1 && i[2] == 0);
}

static void permute(char[] a, int k, Predicate<int[]> decider) {
int n = a.length;
if (k < 1 || k > n)
throw new IllegalArgumentException("Illegal number of positions.");

int[] indexes = new int[n];
int total = (int) Math.pow(n, k);

while (total-- > 0) {
for (int i = 0; i < n - (n - k); i++)
System.out.print(a[indexes[i]]);
System.out.println();

if (decider.test(indexes))
break;

for (int i = 0; i < n; i++) {
if (indexes[i] >= n - 1) {
indexes[i] = 0;
} else {
indexes[i]++;
break;
}
}
}
}
}
```

Output:

```aaa
baa
caa
daa
aba
bba
```

## JavaScript

### ES5

Permutations with repetitions, using strict evaluation, generating the entire set (where system constraints permit) with some degree of efficiency. For lazy or interruptible evaluation, see the second example below.

```(function () {
'use strict';

// permutationsWithRepetition :: Int -> [a] -> [[a]]
var permutationsWithRepetition = function (n, as) {
return as.length > 0 ? (
foldl1(curry(cartesianProduct)(as), replicate(n, as))
) : [];
};

// GENERIC FUNCTIONS -----------------------------------------------------

// cartesianProduct :: [a] -> [b] -> [[a, b]]
var cartesianProduct = function (xs, ys) {
return [].concat.apply([], xs.map(function (x) {
return [].concat.apply([], ys.map(function (y) {
return [
[x].concat(y)
];
}));
}));
};

// foldl1 :: (a -> a -> a) -> [a] -> a
var foldl1 = function (f, xs) {
return xs.length > 0 ? xs.slice(1)
.reduce(f, xs[0]) : [];
};

// replicate :: Int -> a -> [a]
var replicate = function (n, a) {
var v = [a],
o = [];
if (n < 1) return o;
while (n > 1) {
if (n & 1) o = o.concat(v);
n >>= 1;
v = v.concat(v);
}
return o.concat(v);
};

// curry :: ((a, b) -> c) -> a -> b -> c
var curry = function (f) {
return function (a) {
return function (b) {
return f(a, b);
};
};
};

// TEST -----------------------------------------------------------------
// show :: a -> String
var show = function (x) {
return JSON.stringify(x);
}; //, null, 2);

return show(permutationsWithRepetition(2, [1, 2, 3]));

//--> [[1,1],[1,2],[1,3],[2,1],[2,2],[2,3],[3,1],[3,2],[3,3]]
})();
```

{{Out}}

```[[1,1],[1,2],[1,3],[2,1],[2,2],[2,3],[3,1],[3,2],[3,3]]
```

Permutations with repetition by treating the $n^k$ elements as an ordered set, and writing a function from a zero-based index to the nth permutation. This allows us terminate a repeated generation on some condition, or explore a sub-set without needing to generate the whole set:

```(function () {
'use strict';

// nthPermutationWithRepn :: [a] -> Int -> Int -> [a]
var nthPermutationWithRepn = function (xs, groupSize, index) {
var intBase = xs.length,
intSetSize = Math.pow(intBase, groupSize),
lastIndex = intSetSize - 1; // zero-based

if (intBase < 1 || index > lastIndex) return undefined;

var baseElements = unfoldr(function (m) {
var v = m.new,
d = Math.floor(v / intBase);
return {
valid: d > 0,
value: xs[v % intBase],
new: d
};
}, index),
intZeros = groupSize - baseElements.length;

return intZeros > 0 ? replicate(intZeros, xs[0])
.concat(baseElements) : baseElements;
};

// GENERIC FUNCTIONS

// unfoldr :: (b -> Maybe (a, b)) -> b -> [a]
var unfoldr = function (mf, v) {
var xs = [];
return [until(function (m) {
return !m.valid;
}, function (m) {
var m2 = mf(m);
return m2.valid && (xs = [m2.value].concat(xs)), m2;
}, {
valid: true,
value: v,
new: v
})
.value
].concat(xs);
};

// until :: (a -> Bool) -> (a -> a) -> a -> a
var until = function (p, f, x) {
var v = x;
while (!p(v)) {
v = f(v);
}
return v;
};

// replicate :: Int -> a -> [a]
var replicate = function (n, a) {
var v = [a],
o = [];
if (n < 1) return o;
while (n > 1) {
if (n & 1) o = o.concat(v);
n >>= 1;
v = v.concat(v);
}
return o.concat(v);
};

// show :: a -> String
var show = function (x) {
return JSON.stringify(x);
}; //, null, 2);

// curry :: Function -> Function
var curry = function (f) {
for (var lng = arguments.length,
args = Array(lng > 1 ? lng - 1 : 0),
iArg = 1; iArg < lng; iArg++) {
args[iArg - 1] = arguments[iArg];
}

var intArgs = f.length,
go = function (xs) {
return xs.length >= intArgs ? f.apply(null, xs) : function () {
return go(xs.concat([].slice.apply(arguments)));
};
};
return go([].slice.call(args, 1));
};

// range :: Int -> Int -> [Int]
var range = function (m, n) {
return Array.from({
length: Math.floor(n - m) + 1
}, function (_, i) {
return m + i;
});
};

// TEST
// Just items 30 to 35 in the (zero-indexed) series:
return show(range(30, 35)
.map(curry(nthPermutationWithRepn)(['X', 'Y', 'Z'], 4)));
})();
```

{{Out}}

```["Y","X","Y","X"], ["Y","X","Y","Y"], ["Y","X","Y","Z"], ["Y","X","Z","X"], ["Y","X","Z","Y"], ["Y","X","Z","Z"]
```

### =Strict evaluation of the whole set=

Permutations with repetitions, using strict evaluation, generating the entire set. For partial or interruptible evaluation, see the second example below.

A (strict) analogue of the (lazy) replicateM in Haskell.

```(() => {
'use strict';

// GENERIC FUNCTIONS

// replicateM n act performs the action n times, gathering the results.
// replicateM :: (Applicative m) => Int -> m a -> m [a]
const replicateM = (n, f) => {
const loop = x => x <= 0 ? [
[]
] : liftA2(cons, f, loop(x - 1));
return loop(n);
};

// Lift a binary function to actions.
// liftA2 :: Applicative f => (a -> b -> c) -> f a -> f b -> f c
const liftA2 = (f, a, b) =>
listApply(a.map(curry(f)), b);

// <*>
// listApply :: [(a -> b)] -> [a] -> [b]
const listApply = (fs, xs) =>
[].concat.apply([], fs.map(f =>
[].concat.apply([], xs.map(x => [f(x)]))));

// curry :: ((a, b) -> c) -> a -> b -> c
const curry = f => a => b => f(a, b);

// cons :: a -> [a] -> [a]
const cons = (x, xs) => [x].concat(xs);

// show :: a -> String;
const show = JSON.stringify;

// TEST
return show(
replicateM(2, [1, 2, 3])
);
// -> [[1,1],[1,2],[1,3],[2,1],[2,2],[2,3],[3,1],[3,2],[3,3]]
})();
```

{{Out}}

```[[1,1],[1,2],[1,3],[2,1],[2,2],[2,3],[3,1],[3,2],[3,3]]
```

### =Lazy evaluation with a generator =

Permutations with repetition by treating the $n^k$ elements as an ordered set, and writing a function from a zero-based index to the nth permutation. Wrapping this function in a generator allows us terminate a repeated generation on some condition, or explore a sub-set without needing to generate the whole set:

```(() => {
'use strict';

const main = () => {

// Generator object
const gen = permsWithRepn('ACKR', 5);

// Search without needing to generate whole set:
let
nxt = gen.next(),
i = 0,
alpha = nxt.value,
psi = alpha;
while (!nxt.done && 'crack' !== toLower(concat(nxt.value))) {
psi = nxt.value;
console.log(psi)
nxt = gen.next()
i++
}
console.log(nxt.value)
return (
'Generated ' + i + ' of ' + Math.pow(4, 5) +
' possible permutations,\n' +
'searching from: ' + show(alpha) + ' thru: ' + show(psi) +
'\nbefore finding: ' + show(nxt.value)
);
};

// PERMUTATION GENERATOR ------------------------------

// permsWithRepn :: [a] -> Int -> Generator [a]
function* permsWithRepn(xs, intGroup) {
const
vs = Array.from(xs),
intBase = vs.length,
intSet = Math.pow(intBase, intGroup);
if (0 < intBase) {
let index = 0;
while (index < intSet) {
const
ds = unfoldr(
v => 0 < v ? (() => {
const rd = quotRem(v, intBase);
return Just(Tuple(vs[rd[1]], rd[0]))
})() : Nothing(),
index++
);
yield replicate(
intGroup - ds.length,
vs[0]
).concat(ds);
};
}
};

// GENERIC FUNCTIONS ----------------------------------

// Just :: a -> Maybe a
const Just = x => ({
type: 'Maybe',
Nothing: false,
Just: x
});

// Nothing :: Maybe a
const Nothing = () => ({
type: 'Maybe',
Nothing: true,
});

// Tuple (,) :: a -> b -> (a, b)
const Tuple = (a, b) => ({
type: 'Tuple',
'0': a,
'1': b,
length: 2
});

// concat :: [[a]] -> [a]
// concat :: [String] -> String
const concat = xs =>
0 < xs.length ? (() => {
const unit = 'string' !== typeof xs[0] ? (
[]
) : '';
return unit.concat.apply(unit, xs);
})() : [];

// index (!!) :: [a] -> Int -> a
// index (!!) :: String -> Int -> Char
const index = (xs, i) => xs[i];

// quotRem :: Int -> Int -> (Int, Int)
const quotRem = (m, n) =>
Tuple(Math.floor(m / n), m % n);

// replicate :: Int -> a -> [a]
const replicate = (n, x) =>
Array.from({
length: n
}, () => x);

// show :: a -> String
const show = x => JSON.stringify(x);

// toLower :: String -> String
const toLower = s => s.toLocaleLowerCase();

// unfoldr(x => 0 !== x ? Just([x, x - 1]) : Nothing(), 10);
// --> [10,9,8,7,6,5,4,3,2,1]

// unfoldr :: (b -> Maybe (a, b)) -> b -> [a]
const unfoldr = (f, v) => {
let
xr = [v, v],
xs = [];
while (true) {
const mb = f(xr[1]);
if (mb.Nothing) {
return xs
} else {
xr = mb.Just;
xs.push(xr[0])
}
}
};

// MAIN ---
return main();
})();
```

{{Out}}

```Generated 589 of 1024 possible permutations,
searching from: ["A","A","A","A","A"] thru: ["A","R","A","C","K"]
before finding: ["C","R","A","C","K"]
```

## jq

We first present a definition of permutations_with_replacement(n) that is compatible with jq 1.4. To interrupt the stream that it produces, however, requires a version of jq with break, which was introduced after the release of jq 1.4.

'''Definitions'''

We shall define permutations_with_replacements(n) in terms of a more general filter, combinations/0, defined as follows:

```# Input: an array, \$in, of 0 or more arrays
# Output: a stream of arrays, c, with c[i] drawn from \$in[i].
def combinations:
if length == 0 then []
else
.[0][] as \$x
| (.[1:] | combinations) as \$y
| [\$x] +  \$y
end ;

# Input: an array of the k values from which to choose.
# Output: a stream of arrays of length n with elements drawn from the input array.
def permutations_with_replacements(n):
. as \$in | [range(0; n) | \$in] | combinations;
```

'''Example 1: Enumeration''':

Count the number of 4-combinations of [0,1,2] by enumerating them, i.e., without creating a data structure to store them all.

```def count(stream): reduce stream as \$i (0; .+1);

count([0,1,2] | permutations_with_replacements(4))
# output: 81
```

'''Example 2: Early termination of the generator''':

Counting from 1, and terminating the generator when the item is found, what is the sequence number of ["c", "a", "b"] in the stream of 3-combinations of ["a","b","c"]?

```# Input: the item to be matched
# Output: the index of the item in the stream (counting from 1);
# emit null if the item is not found
def sequence_number(stream):
. as \$in
| (label \$top
| foreach stream as \$i (0; .+1; if \$in == \$i then ., break \$top else empty end))
// null;  # NOTE: "//" here is an operator

["c", "a", "b"] | sequence_number( ["a","b","c"] | permutations_with_replacements(3))

# output: 20
```

## Julia

{{works with|Julia|0.6}}

Implements a simil-Combinatorics.jl API.

```struct WithRepetitionsPermutations{T}
a::T
t::Int
end

with_repetitions_permutations(elements::T, len::Integer) where T =
WithRepetitionsPermutations{T}(unique(elements), len)

Base.iteratorsize(::WithRepetitionsPermutations) = Base.HasLength()
Base.length(p::WithRepetitionsPermutations) = length(p.a) ^ p.t
Base.iteratoreltype(::WithRepetitionsPermutations) = Base.HasEltype()
Base.eltype(::WithRepetitionsPermutations{T}) where T = T
Base.start(p::WithRepetitionsPermutations) = ones(Int, p.t)
Base.done(p::WithRepetitionsPermutations, s::Vector{Int}) = s[end] > endof(p.a)
function Base.next(p::WithRepetitionsPermutations, s::Vector{Int})
cur = p.a[s]
s[1] += 1
local i = 1
while i < endof(s) && s[i] > length(p.a)
s[i] = 1
s[i+1] += 1
i += 1
end
return cur, s
end

println("Permutations of [4, 5, 6] in 3:")
foreach(println, collect(with_repetitions_permutations([4, 5, 6], 3)))
```

{{out}}

```Permutations of [4, 5, 6] in 3:
[4, 4, 4]
[5, 4, 4]
[6, 4, 4]
[4, 5, 4]
[5, 5, 4]
[6, 5, 4]
[4, 6, 4]
[5, 6, 4]
[6, 6, 4]
[4, 4, 5]
[5, 4, 5]
[6, 4, 5]
[4, 5, 5]
[5, 5, 5]
[6, 5, 5]
[4, 6, 5]
[5, 6, 5]
[6, 6, 5]
[4, 4, 6]
[5, 4, 6]
[6, 4, 6]
[4, 5, 6]
[5, 5, 6]
[6, 5, 6]
[4, 6, 6]
[5, 6, 6]
[6, 6, 6]
```

## K

enlist each from x on the left and each from x on the right where x is range 10

```
,/x/:\:x:!10

```

## Kotlin

{{trans|Go}}

```// version 1.1.2

fun main(args: Array<String>) {
val n  = 3
val values = charArrayOf('A', 'B', 'C', 'D')
val k = values.size
// terminate when first two characters of the permutation are 'B' and 'C' respectively
val decide = fun(pc: CharArray) = pc[0] == 'B' && pc[1] == 'C'
val pn = IntArray(n)
val pc = CharArray(n)
while (true) {
// generate permutation
for ((i, x) in pn.withIndex()) pc[i] = values[x]
// show progress
println(pc.contentToString())
// pass to deciding function
if (decide(pc)) return  // terminate early
// increment permutation number
var i = 0
while (true) {
pn[i]++
if (pn[i] < k) break
pn[i++] = 0
if (i == n) return  // all permutations generated
}
}
}
```

{{out}}

```
[A, A, A]
[B, A, A]
[C, A, A]
[D, A, A]
[A, B, A]
[B, B, A]
[C, B, A]
[D, B, A]
[A, C, A]
[B, C, A]

```

## M2000 Interpreter

```
Module Checkit {
a=("A","B","C","D")
n=len(a)
c1=lambda a, n, c (&f) ->{
=(array(a, c),)
c++
if c=n then c=0: f=true
}
m=n-2
While m >0 {
c3=lambda c2=c1, a, n, c (&f) -> {
f=false
=Cons((array(a, c),), c2(&f))
if f then {
c++
f=false
if c=n then c=0: f=true
}
}
c1=c3
m--
}
k=false
While not k {
r=c3(&k)
rr=each(r end to start)
While rr {
Print array\$(rr),
}
Print
if array\$(r, 2)="B" and array\$(r,1)="C" then exit
}
}
Checkit

```

{{out}}

```A   A   A
B   A   A
C   A   A
D   A   A
A   B   A
B   B   A
C   B   A
D   B   A
A   C   A
B   C   A
```
## Mathematica ```mathematica Tuples[{1, 2, 3}, 2] ``` {{out}} ```txt {{1, 1}, {1, 2}, {1, 3}, {2, 1}, {2, 2}, {2, 3}, {3, 1}, {3, 2}, {3, 3}} ``` ## Maxima ```maxima apply(cartesian_product,makelist({1,2,3}, 2)); ``` {{out}} ```txt {[1,1],[1,2],[1,3],[2,1],[2,2],[2,3],[3,1],[3,2],[3,3]} ``` ## Perl ```perl use Algorithm::Combinatorics qw/tuples_with_repetition/; print join(" ", map { "[@\$_]" } tuples_with_repetition([qw/A B C/],2)), "\n"; ``` {{out}} ```txt [A A] [A B] [A C] [B A] [B B] [B C] [C A] [C B] [C C] ``` Solving the crack problem: ```perl use Algorithm::Combinatorics qw/tuples_with_repetition/; my \$iter = tuples_with_repetition([qw/A C K R/], 5); my \$tries = 0; while (my \$p = \$iter->next) { \$tries++; die "Found the combination after \$tries tries!\n" if join("",@\$p) eq "CRACK"; } ``` {{out}} ```txt Found the combination after 455 tries! ``` ## Perl 6 We can use the X operator ("cartesian product") to cross the list with itself. For $n=2$: {{works with|rakudo|2016.07}} ```perl6>my @k = n: {{works with|rakudo|2016.07}} ```perl6>my @k = k^n possibilities in base $k$: {{works with|rakudo|2016.07}} ```perl6>my @k = 0 then Begin mdTup_k:= k; setlength(mdTup,k); IF (n<0) then mdTup_n := 0 else mdTup_n := n; end else Begin mdTup_k := 1; mdTup_n := k; end; end; end; procedure PermOut(const p:tPermData); var i : nativeInt; Begin with p do Begin For i := 0 to mdTup_k-1 do write(mdTup[i]:4); end; writeln; end; function NextPermWithRep(var perm:tPermData): boolean; // create next permutation by adding 1 and correct "carry" // returns false if finished var pDg :^Integer; dg,le :nativeInt; begin WIth perm do Begin pDg := @mdTup[0]; le := mdTup_k; repeat dg := pDg^+1; IF (dg 8^10 real 0m2.556s // without inc(cnt); real 0m2.288s-> 7,5 cycles per call //"old" compiler-version //real 0m3.465s /fpc/2.6.4/ppc386 "%f" -al -Xs -XX -O3 ``` ## Phix The task is equivalent to simply counting in base=length(set), from 1 to power(base,n). Asking for the 0th permutation just returns the total number of permutations (ie ""). Results can be generated in any order, hence early termination is quite simply a non-issue. ```Phix function permrep(sequence set, integer n, idx=0) integer base = length(set), nperm = power(base,n) if idx=0 then -- return the number of permutations return nperm end if -- return the idx'th [1-based] permutation if idx<1 or idx>nperm then ?9/0 end if idx -= 1 -- make it 0-based sequence res = "" for i=1 to n do res = prepend(res,set[mod(idx,base)+1]) idx = floor(idx/base) end for if idx!=0 then ?9/0 end if -- sanity check return res end function ``` Some slightly excessive testing: ```Phix procedure show_all(sequence set, integer n) integer l = permrep(set,n) sequence s = repeat(0,l) for i=1 to l do s[i] = permrep(set,n,i) end for ?s end procedure show_all("123",1) show_all("123",2) show_all("123",3) show_all("456",3) show_all({1,2,3},3) show_all({"bat","fox","cow"},2) sequence s = {} for i=31 to 36 do s = append(s,permrep("XYZ",4,i)) end for ?s integer l = permrep("ACKR",5) for i=1 to l do if permrep("ACKR",5,i)="CRACK" then -- 455 printf(1,"Permutation %d of %d: CRACK\n",{i,l}) exit end if end for --The 590th (one-based) permrep is KCARC, ie reverse(CRACK), matching the 589 result of 0-based idx solutions printf(1,"reverse(permrep(\"ACKR\",5,589+1):%s\n",{reverse(permrep("ACKR",5,590))}) ``` {{out}} ```txt {"1","2","3"} {"11","12","13","21","22","23","31","32","33"} {"111","112","113","121","122","123","131","132","133","211","212","213","221","222","223","231","232","233","311","312","313","321","322","323","331","332","333"} {"444","445","446","454","455","456","464","465","466","544","545","546","554","555","556","564","565","566","644","645","646","654","655","656","664","665","666"} {{1,1,1},{1,1,2},{1,1,3},{1,2,1},{1,2,2},{1,2,3},{1,3,1},{1,3,2},{1,3,3},{2,1,1},{2,1,2},{2,1,3},{2,2,1},{2,2,2},{2,2,3},{2,3,1},{2,3,2},{2,3,3},{3,1,1},{3,1,2},{3,1,3},{3,2,1},{3,2,2},{3,2,3},{3,3,1},{3,3,2},{3,3,3}} {{"bat","bat"},{"bat","fox"},{"bat","cow"},{"fox","bat"},{"fox","fox"},{"fox","cow"},{"cow","bat"},{"cow","fox"},{"cow","cow"}} {"YXYX","YXYY","YXYZ","YXZX","YXZY","YXZZ"} Permutation 455 of 1024: CRACK reverse(permrep("ACKR",5,589+1):CRACK ``` ## PHP ```PHP Int -> m a -> m [a] def replicateM(n): '''A functor collecting values accumulated by n repetitions of m. (List instance only here). ''' def rep(m): def go(x): return [[]] if 1 > x else ( liftA2List(lambda a, b: [a] + b)(m)(go(x - 1)) ) return go(n) return lambda m: rep(m) # TEST ---------------------------------------------------- # main :: IO () def main(): '''Permutations of two elements, drawn from three values''' print( fTable(main.__doc__ + ':\n')(repr)(showList)( replicateM(2) )([[1, 2, 3], 'abc']) ) # GENERIC FUNCTIONS --------------------------------------- # liftA2List :: (a -> b -> c) -> [a] -> [b] -> [c] def liftA2List(f): '''The binary operator f lifted to a function over two lists. f applied to each pair of arguments in the cartesian product of xs and ys. ''' return lambda xs: lambda ys: [ f(*xy) for xy in product(xs, ys) ] # DISPLAY ------------------------------------------------- # fTable :: String -> (a -> String) -> # (b -> String) -> (a -> b) -> [a] -> String def fTable(s): '''Heading -> x display function -> fx display function -> f -> xs -> tabular string. ''' def go(xShow, fxShow, f, xs): ys = [xShow(x) for x in xs] w = max(map(len, ys)) return s + '\n' + '\n'.join(map( lambda x, y: y.rjust(w, ' ') + ' -> ' + fxShow(f(x)), xs, ys )) return lambda xShow: lambda fxShow: lambda f: lambda xs: go( xShow, fxShow, f, xs ) # showList :: [a] -> String def showList(xs): '''Stringification of a list.''' return '[' + ','.join( showList(x) if isinstance(x, list) else repr(x) for x in xs ) + ']' # MAIN --- if __name__ == '__main__': main() ``` {{Out}} ```txt Permutations of two elements, drawn from three values: [1, 2, 3] -> [[1,1],[1,2],[1,3],[2,1],[2,2],[2,3],[3,1],[3,2],[3,3]] 'abc' -> [['a','a'],['a','b'],['a','c'],['b','a'],['b','b'],['b','c'],['c','a'],['c','b'],['c','c']] ``` ### Lazy evaluation with a generator ### =Applying itertools.product= ```python from itertools import product # check permutations until we find the word 'crack' for x in product('ACRK', repeat=5): w = ''.join(x) print w if w.lower() == 'crack': break ``` ### =Writing a generator= Or, composing our own generator, by wrapping a function '''from''' an index in the range ''0 .. ((distinct items to the power of groupSize) - 1)'' '''to''' a unique permutation. (Each permutation is equivalent to a 'number' in the base of the size of the set of distinct items, in which each distinct item functions as a 'digit'): {{Works with|Python|3.7}} ```Python '''Generator-based permutations with repetition''' from itertools import (chain, repeat) # permsWithRepns :: [a] -> Int -> Generator [[a]] def permsWithRepns(xs): '''Generator of permutations of length n, with elements drawn from the values in xs. ''' def groupsOfSize(n): f = nthPermWithRepn(xs)(n) limit = len(xs)**n i = 0 while i < limit: yield f(i) i = 1 + i return lambda n: groupsOfSize(n) # Index as a 'number' in the base of the # size of the set (of distinct values to be permuted), # using each value as a 'digit' # (leftmost value used as the 'zero') # nthPermWithRepn :: [a] -> Int -> Int -> [a] def nthPermWithRepn(xs): '''Indexed permutation of n values drawn from xs''' def go(intGroup, index): vs = list(xs) intBase = len(vs) intSet = intBase ** intGroup return ( lambda ds=unfoldr( lambda v: ( lambda qr=divmod(v, intBase): Just( (qr[0], vs[qr[1]]) ) )() if 0 < v else Nothing() )(index): ( list(repeat(vs[0], intGroup - len(ds))) + ds ) )() if 0 < intBase and index < intSet else None return lambda intGroup: lambda index: go( intGroup, index ) # MAIN ---------------------------------------------------- # main :: IO () def main(): '''Search for a 5 char permutation drawn from 'ACKR' matching "crack"''' cs = 'ACKR' wordLength = 5 target = 'crack' gen = permsWithRepns(cs)(wordLength) mb = Nothing() for idx, xs in enumerate(gen): s = ''.join(xs) if target == s.lower(): mb = Just((s, idx)) break print(main.__doc__ + ':\n') print( maybe('No match found for "{k}"'.format(k=target))( lambda m: 'Permutation {idx} of {total}: {pm}'.format( idx=m[1], total=len(cs)**wordLength, pm=s ) )(mb) ) # GENERIC FUNCTIONS ------------------------------------- # Just :: a -> Maybe a def Just(x): '''Constructor for an inhabited Maybe(option type) value.''' return {'type': 'Maybe', 'Nothing': False, 'Just': x} # Nothing :: Maybe a def Nothing(): '''Constructor for an empty Maybe(option type) value.''' return {'type': 'Maybe', 'Nothing': True} # concat :: [[a]] -> [a] # concat :: [String] -> String def concat(xs): '''The concatenation of all the elements in a list or iterable.''' def f(ys): zs = list(chain(*ys)) return ''.join(zs) if isinstance(ys[0], str) else zs return ( f(xs) if isinstance(xs, list) else ( chain.from_iterable(xs) ) ) if xs else [] # fst :: (a, b) -> a def fst(tpl): '''First member of a pair.''' return tpl[0] # maybe :: b -> (a -> b) -> Maybe a -> b def maybe(v): '''Either the default value v, if m is Nothing, or the application of f to x, where m is Just(x). ''' return lambda f: lambda m: v if None is m or m.get('Nothing') else ( f(m.get('Just')) ) # snd :: (a, b) -> b def snd(tpl): '''Second member of a pair.''' return tpl[1] # unfoldr(lambda x: Just((x, x - 1)) if 0 != x else Nothing())(10) # -> [10, 9, 8, 7, 6, 5, 4, 3, 2, 1] # unfoldr :: (b -> Maybe (a, b)) -> b -> [a] def unfoldr(f): '''Dual to reduce or foldr. Where catamorphism reduces a list to a summary value, the anamorphic unfoldr builds a list from a seed value. As long as f returns Just(a, b), a is prepended to the list, and the residual b is used as the argument for the next application of f. When f returns Nothing, the completed list is returned. ''' def go(v): xr = v, v xs = [] while True: mb = f(xr[0]) if mb.get('Nothing'): return xs else: xr = mb.get('Just') xs.append(xr[1]) return xs return lambda x: go(x) # MAIN --- if __name__ == '__main__': main() ``` {{Out}} ```txt Search for a 5 char permutation drawn from 'ACKR' matching "crack": Permutation 589 of 1024: CRACK ``` ## Racket ### As a sequence First we define a procedure that defines the sequence of the permutations. ```Racket #lang racket (define (permutations-with-repetitions/proc size items) (define items-vector (list->vector items)) (define num (length items)) (define (pos->element pos) (reverse (for/list ([p (in-vector pos)]) (vector-ref items-vector p)))) (define (next-pos pos) (let ([ret (make-vector size #f)]) (for/fold ([carry 1]) ((i (in-range size))) (let ([tmp (+ (vector-ref pos i) carry)]) (if (= tmp num) (begin (vector-set! ret i 0) #;carry 1) (begin (vector-set! ret i tmp) #;carry 0)))) ret)) (define initial-pos (vector->immutable-vector (make-vector size 0))) (define last-pos (vector->immutable-vector (make-vector size (sub1 num)))) (define (continue-after-pos+val? pos val) (not (equal? pos last-pos))) (make-do-sequence (lambda () (values pos->element next-pos initial-pos #f #f continue-after-pos+val?)))) (sequence->list (permutations-with-repetitions/proc 2 '(1 2 3))) ``` {{out}} ```txt '((1 1) (1 2) (1 3) (2 1) (2 2) (2 3) (3 1) (3 2) (3 3)) ``` ### As a sequence with for clause support Now we define a more general version that can be used efficiently in as a for clause. In other uses it falls back to the sequence implementation. ```Racket (require (for-syntax racket)) (define-sequence-syntax in-permutations-with-repetitions (lambda () #'permutations-with-repetitions/proc) (lambda (stx) (syntax-case stx () [[(element) (_ size/ex items/ex)] #'[(element) (:do-in ([(size) size/ex] [(items) items/ex] [(items-vector) (list->vector items/ex)] [(num) (length items/ex)] [(last-pos) (make-vector size/ex (sub1 (length items/ex)))]) (void) ([pos (make-vector size 0)]) #t ([(element) (reverse (for/list ([p (in-vector pos)]) (vector-ref items-vector p)))]) #t (not (equal? pos last-pos)) [(let ([ret (make-vector size #f)]) (for/fold ([carry 1]) ((i (in-range size))) (let ([tmp (+ (vector-ref pos i) carry)]) (if (= tmp num) (begin (vector-set! ret i 0) #;carry 1) (begin (vector-set! ret i tmp) #;carry 0)))) ret)])]]))) (for/list ([element (in-permutations-with-repetitions 2 '(1 2 3))]) element) (sequence->list (in-permutations-with-repetitions 2 '(1 2 3))) ``` {{out}} ```txt '((1 1) (1 2) (1 3) (2 1) (2 2) (2 3) (3 1) (3 2) (3 3)) '((1 1) (1 2) (1 3) (2 1) (2 2) (2 3) (3 1) (3 2) (3 3)) ``` ## REXX ### version 1 ```rexx /*REXX pgm generates/displays all permutations of N different objects taken M at a time.*/ parse arg things bunch inbetweenChars names /* ╔════════════════════════════════════════════════════════════════╗ */ /* ║ inBetweenChars (optional) defaults to a [null]. ║ */ /* ║ names (optional) defaults to digits (and letters).║ */ /* ╚════════════════════════════════════════════════════════════════╝ */ call permSets things, bunch, inBetweenChars, names exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ p: return word( arg(1), 1) /*P function (Pick first arg of many).*/ /*──────────────────────────────────────────────────────────────────────────────────────*/ permSets: procedure; parse arg x,y,between,uSyms /*X things taken Y at a time. */ @.=; sep= /*X can't be > length(@0abcs). */ @abc = 'abcdefghijklmnopqrstuvwxyz'; @abcU= @abc; upper @abcU @abcS = @abcU || @abc; @0abcS= 123456789 || @abcS do k=1 for x /*build a list of permutation symbols. */ _= p( word(uSyms, k) p( substr(@0abcS, k, 1) k) ) /*get/generate a symbol.*/ if length(_)\==1 then sep= '_' /*if not 1st character, then use sep. */ \$.k= _ /*append the character to symbol list. */ end /*k*/ if between=='' then between= sep /*use the appropriate separator chars. */ call .permSet 1 /*start with the first permutation. */ return /* [↓] this is a recursive subroutine.*/ .permSet: procedure expose \$. @. between x y; parse arg ? if ?>y then do; _=@.1; do j=2 for y-1; _=_ || between || @.j; end; say _ end else do q=1 for x /*build the permutation recursively. */ @.?= \$.q; call .permSet ?+1 end /*q*/ return /*this is meant to be an anonymous sub.*/ ``` {{out|output|text= when using the default inputs of: 3 2 }} ```txt 11 12 13 21 22 23 31 32 33 ``` {{out|output|text= when using the default inputs of : 3 2 , bat fox cow }} ```txt bat,bat bat,fox bat,cow fox,bat fox,fox fox,cow cow,bat cow,fox cow,cow ``` ===version 2 (using Interpret)=== Note: this REXX version will cause Regina REXX to fail (crash) if the expression to be INTERPRETed is too large (byte-wise). PC/REXX and Personal REXX also fail, but for a smaller expression. Please specify limitations. One could add: If length(a)>implementation_dependent_limit Then Say 'too large for this Rexx version' Also note that the output isn't the same as REXX version 1 when the 1st argument is two digits or more, i.e.: '''11 2''' ```rexx /* REXX *************************************************************** * Arguments and output as in REXX version 1 (for the samples shown there) * For other elements (such as 11 2), please specify a separator * Translating 10, 11, etc. to A, B etc. is left to the reader * 12.05.2013 Walter Pachl * 12-05-2013 Walter Pachl take care of bunch<=0 and other oddities **********************************************************************/ Parse Arg things bunch sep names If datatype(things,'W') & datatype(bunch,'W') Then Nop Else Call exit 'First two arguments must be integers >0' If things='' Then n=3; Else n=things If bunch='' Then m=2; Else m=bunch If things<=0 Then Call exit 'specify a positive number of things' If bunch<=0 Then Call exit 'no permutations with' bunch 'elements!' Select When sep='' Then ss='''''' When datatype(sep)='NUM' Then ss=''''copies(' ',sep)'''' Otherwise ss=''''sep'''' End Do i=1 To n If names<>'' Then Parse Var names e.i names Else e.i=i End a='p=0;'; Do i=1 To m; a=a||'Do p'i'=1 To n;'; End a=a||'ol=e.p1' Do i=2 To m; a=a||'||'ss'||e.p'i; End a=a||'; say ol; p=p+1;' Do i=1 To m; a=a||'end;'; End a=a||'Say' p 'permutations' /* Say a */ Interpret a ``` ### version 3 This is a very simplistic version that is limited to nine things ('''N'''). It essentially just executes a '''do''' loop and ignores any permutation out of range, this is very wasteful of CPU processing time when using a larger '''N'''. This version could easily be extended to '''N''' up to 15 (using hexadecimal arithmetic). ```rexx /*REXX pgm gens all permutations with repeats of N objects (<10) taken M at a time. */ parse arg N M . z= N**M \$= left(1234567890, N) t= 0 do j=copies(1, M) until t==z if verify(j, \$)\==0 then iterate t= t+1 say j end /*j*/ /*stick a fork in it, we're all done. */ ``` {{out|output|text= when using the following inputs: 3 2 }} ```txt 11 12 13 21 22 23 31 32 33 ``` ## Ring ```ring # Project : Permutations with repetitions list1 = [["a", "b", "c"], ["a", "b", "c"]] list2 = [["1", "2", "3"], ["1", "2", "3"]] permutation(list1) permutation(list2) func permutation(list1) for n = 1 to len(list1[1]) for m = 1 to len(list1[2]) see list1[1][n] + " " + list1[2][m] + nl next next see nl ``` Output: ```txt a a a b a c b a b b b c c a c b c c 1 1 1 2 1 3 2 1 2 2 2 3 3 1 3 2 3 3 ``` ## Ruby This is built in　(Array#repeated_permutation): ```ruby rp = [1,2,3].repeated_permutation(2) # an enumerator (generator) p rp.to_a #=>[[1, 1], [1, 2], [1, 3], [2, 1], [2, 2], [2, 3], [3, 1], [3, 2], [3, 3]] #yield permutations until their sum happens to exceed 4, then quit: p rp.take_while{|(a, b)| a + b < 5} #=>[[1, 1], [1, 2], [1, 3], [2, 1], [2, 2]] ``` ## Scala ```scala package permutationsRep object PermutationsRepTest extends Application { /** * Calculates all permutations taking n elements of the input List, * with repetitions. * Precondition: input.length > 0 && n > 0 */ def permutationsWithRepetitions[T](input : List[T], n : Int) : List[List[T]] = { require(input.length > 0 && n > 0) n match { case 1 => for (el <- input) yield List(el) case _ => for (el <- input; perm <- permutationsWithRepetitions(input, n - 1)) yield el :: perm } } println(permutationsWithRepetitions(List(1, 2, 3), 2)) } ``` {{out}} ```txt List(List(1, 1), List(1, 2), List(1, 3), List(2, 1), List(2, 2), List(2, 3), List(3, 1), List(3, 2), List(3, 3)) ``` ## Sidef ```ruby var k = %w(a b c) var n = 2 cartesian([k] * n, {|*a| say a.join(' ') }) ``` {{out}} ```txt a a a b a c b a b b b c c a c b c c ``` ## Tcl ### Iterative version {{trans|PHP}} ```tcl proc permutate {values size offset} { set count [llength \$values] set arr [list] for {set i 0} {\$i < \$size} {incr i} { set selector [expr [round [expr \$offset / [pow \$count \$i]]] % \$count]; lappend arr [lindex \$values \$selector] } return \$arr } proc permutations {values size} { set a [list] set c [pow [llength \$values] \$size] for {set i 0} {\$i < \$c} {incr i} { set permutation [permutate \$values \$size \$i] lappend a \$permutation } return \$a } # Usage permutations [list 1 2 3 4] 3 ``` ### Version without additional libraries {{works with|Tcl|8.6}} {{trans|Scala}} ```tcl package require Tcl 8.6 # Utility function to make procedures that define generators proc generator {name arguments body} { set body [list try \$body on ok {} {return -code break}] set lambda [list \$arguments "yield \[info coroutine\];\$body"] proc \$name args "tailcall \ coroutine gen_\[incr ::generate_ctr\] apply [list \$lambda] {*}\\$args" } # How to generate permutations with repetitions generator permutationsWithRepetitions {input n} { if {[llength \$input] == 0 || \$n < 1} {error "bad arguments"} if {![incr n -1]} { foreach el \$input { yield [list \$el] } } else { foreach el \$input { set g [permutationsWithRepetitions \$input \$n] while 1 { yield [list \$el {*}[\$g]] } } } } # Demonstrate usage set g [permutationsWithRepetitions {1 2 3} 2] while 1 {puts [\$g]} ``` ### Alternate version with extra library package {{tcllib|generator}} {{works with|Tcl|8.6}} ```tcl package require Tcl 8.6 package require generator # How to generate permutations with repetitions generator define permutationsWithRepetitions {input n} { if {[llength \$input] == 0 || \$n < 1} {error "bad arguments"} if {![incr n -1]} { foreach el \$input { generator yield [list \$el] } } else { foreach el \$input { set g [permutationsWithRepetitions \$input \$n] while 1 { generator yield [list \$el {*}[\$g]] } } } } # Demonstrate usage generator foreach val [permutationsWithRepetitions {1 2 3} 2] { puts \$val } ```