Permutations with repetitions
Permutations with repetitions is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
- Task
Generate a sequence of permutations of n elements drawn from choice of k values.
This sequence will have 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.
See Also:
The number of samples of size k from n objects.
With combinations and permutations generation tasks.
Order Unimportant Order Important Without replacement Task: Combinations Task: Permutations With replacement Task: Combinations with repetitions Task: Permutations with repetitions
11l
V n = 3
V values = [‘A’, ‘B’, ‘C’, ‘D’]
V k = values.len
V decide = pc -> pc[0] == ‘B’ & pc[1] == ‘C’
V pn = [0] * n
V pc = ["\0"] * n
L
L(x) pn
pc[L.index] = values[x]
print(pc)
I decide(pc)
L.break
V i = 0
L
pn[i]++
I pn[i] < k
L.break
pn[i] = 0
i++
I i == n
^L.break- Output:
[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]
ALGOL 68
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
);
SKIPFile: 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
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
- Output:
{{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 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"
use scripting additions
-- 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
- Output:
Permutation 589 of 1024: CRACK Found after searching from AAAAA thru ARACK
Arturo
decide: function [pc]->
and? pc\0 = `B`
pc\1 = `C`
permutation: function [vals, n][
k: size vals
pn: array.of:n 0
p: array.of:n `0`
while [true][
loop.with:'i pn 'x -> p\[i]: vals\[x]
print p
if decide p -> return ø
i: 0
while [true][
pn\[i]: pn\[i] + 1
if pn\[i] < k -> break
pn\[i]: 0
i: i + 1
if i = n -> return ø
]
]
]
permutation "ABCD" 3
- Output:
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
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
}
AWK
# syntax: GAWK -f PERMUTATIONS_WITH_REPETITIONS.AWK
# converted from C
BEGIN {
numbers = 3
upto = 4
for (tmp2=1; tmp2<=numbers; tmp2++) {
arr[tmp2] = 1
}
arr[numbers] = 0
tmp1 = numbers
while (1) {
if (arr[tmp1] == upto) {
if (--tmp1 == 0) {
break
}
}
else {
arr[tmp1]++
while (tmp1 < numbers) {
arr[++tmp1] = 1
}
printf("(")
for (tmp2=1; tmp2<=numbers; tmp2++) {
printf("%d",arr[tmp2])
}
printf(")")
}
}
printf("\n")
exit(0)
}
- Output:
(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)
BASIC
QBasic
DIM list1$(1 TO 2, 1 TO 3) '= {{"a", "b", "c"}, {"a", "b", "c"}}
DIM list2$(1 TO 2, 1 TO 3) '= {{"1", "2", "3"}, {"1", "2", "3"}}
permutation$(list1$())
PRINT
permutation$(list2$())
END
SUB permutation$(list1$())
FOR n = 1 TO UBOUND(list1$,1)
FOR m = 1 TO UBOUND(list1$,2)
PRINT list1$(1, n); " "; list1$(2, m)
NEXT m
NEXT n
PRINT
END SUB
- Output:
Same as FreeBASIC entry.
BASIC256
arraybase 1
dim list1 = {{"a", "b", "c"}, {"a", "b", "c"}}
dim list2 = {{"1", "2", "3"}, {"1", "2", "3"}}
call permutation(list1)
print
call permutation(list2)
end
subroutine permutation(list1)
for n = 1 to list1[][?]
for m = 1 to list1[][?]
print list1[1, n]; " "; list1[2, m]
next m
next n
print
end subroutine- Output:
Same as FreeBASIC entry.
FreeBASIC
Dim As String list1(1 To 2, 1 To 3) = {{"a", "b", "c"}, {"a", "b", "c"}}
Dim As String list2(1 To 2, 1 To 3) = {{"1", "2", "3"}, {"1", "2", "3"}}
Sub permutation(list() As String)
Dim As Integer n, m
For n = Lbound(list,2) To Ubound(list,2)
For m = Lbound(list,2) To Ubound(list,2)
Print list(1, n); " "; list(2, m)
Next m
Next n
Print
End Sub
permutation(list1())
Print
permutation(list2())
Sleep
- Output:
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
C
#include <stdio.h>
#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;
}
- Output:
(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;
}
- Output:
(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
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;
}
- Output:
[[1, 1], [1, 2], [1, 3], [2, 1], [2, 2], [2, 3], [3, 1], [3, 2], [3, 3]]
Generator Range Version
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.
EasyLang
n = 3
values$[] = strchars "ABCD"
#
func decide pc$[] .
return if pc$[1] = "B" and pc$[2] = "C"
.
for i to n
pn[] &= 1
.
len pc$[] n
repeat
for i to len pn[]
pc$[i] = values$[pn[i]]
.
print strjoin pc$[]
until decide pc$[] = 1
i = 1
repeat
pn[i] += 1
until pn[i] <= len values$[]
pn[i] = 1
i += 1
if i = n
print "done"
break 2
.
.
.
- Output:
AAA BAA CAA DAA ABA BBA CBA DBA ACA BCA
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
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)
- Output:
[[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
}
}
}
}
- Output:
[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]
Haskell
import Control.Monad (replicateM)
main = mapM_ print (replicateM 2 [1,2,3])
- Output:
[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
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]]
})();
- Output:
[[1,1],[1,2],[1,3],[2,1],[2,2],[2,3],[3,1],[3,2],[3,3]]
Permutations with repetition by treating the 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)));
})();
- Output:
["Y","X","Y","X"], ["Y","X","Y","Y"], ["Y","X","Y","Z"], ["Y","X","Z","X"], ["Y","X","Z","Y"], ["Y","X","Z","Z"]
ES6
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]]
})();
- Output:
[[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 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();
})();
- Output:
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: 20Julia
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)))
- Output:
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
// 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
}
}
}
- Output:
[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- Output:
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 /Wolfram Language
Tuples[{1, 2, 3}, 2]
- Output:
{{1, 1}, {1, 2}, {1, 3}, {2, 1}, {2, 2}, {2, 3}, {3, 1}, {3, 2}, {3, 3}}
Maxima
apply(cartesian_product,makelist({1,2,3}, 2));
- Output:
{[1,1],[1,2],[1,3],[2,1],[2,2],[2,3],[3,1],[3,2],[3,3]}
Nim
import strutils
func decide(pc: openArray[char]): bool =
## Terminate when first two characters of the
## permutation are 'B' and 'C' respectively.
pc[0] == 'B' and pc[1] == 'C'
proc permute(values: openArray[char]; n: Positive) =
let k = values.len
var
pn = newSeq[int](n)
p = newSeq[char](n)
while true:
# Generate permutation
for i, x in pn: p[i] = values[x]
# Show progress.
echo p.join(" ")
# Pass to deciding function.
if decide(p): return # Terminate early.
# Increment permutation number.
var i = 0
while true:
inc pn[i]
if pn[i] < k: break
pn[i] = 0
inc i
if i == n: return # All permutations generated.
permute("ABCD", 3)
- Output:
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
Pascal
Create a list of indices into what ever you want, one by one. Doing it by addig one to a number with k-positions to base n.
program PermuWithRep;
//permutations with repetitions
//http://rosettacode.org/wiki/Permutations_with_repetitions
{$IFDEF FPC}
{$Mode Delphi}{$Optimization ON}{$Align 16}{$Codealign proc=16,loop=4}
{$ELSE}
{$APPTYPE CONSOLE}// for Delphi
{$ENDIF}
uses
sysutils;
type
tPermData = record
mdTup_n, //number of positions
mdTup_k:NativeInt; //number of different elements
mdTup :array of integer;
end;
function InitTuple(k,n:nativeInt):tPermData;
begin
with result do
Begin
IF 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<mdTup_n) then
Begin
pDg^ := dg;
BREAK;
end
else
pDg^ := 0;
dec(le);
inc(pDg);
until le<=0;
result := (dg<mdTup_n);
end;
end;
var
p: tPermData;
cnt,k,n: nativeInt;
Begin
cnt := 0;
//k := 2;n := 3;
k := 10;n := 8;
p:= InitTuple(k,n);
IF (n<= 6) then
repeat
inc(cnt);
PermOut(p);
until Not(NextPermWithRep(p))
else
repeat
inc(cnt);
until Not(NextPermWithRep(p));
writeln('k: ',k,' n: ',n,' count ',cnt);
end.
- Output:
0 0 1 0 2 0 0 1 1 1 2 1 0 2 1 2 2 2 k: 2 n: 3 count 9 .. //speedtest Compiler /fpc/3.1.1/ppc386 "%f" -al -Xs -XX -O3 // i4330 3.5 Ghz k: 10 n: 8 count 1073741824 => 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
Perl
use Algorithm::Combinatorics qw/tuples_with_repetition/;
print join(" ", map { "[@$_]" } tuples_with_repetition([qw/A B C/],2)), "\n";
- Output:
[A A] [A B] [A C] [B A] [B B] [B C] [C A] [C B] [C C]
Solving the crack problem:
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";
}
- Output:
Found the combination after 455 tries!
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.
with javascript_semantics 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: procedure show_all(sequence set, integer n, lo=1, hi=0) integer l = permrep(set,n) if hi=0 then hi=l end if sequence s = repeat(0,l) for i=1 to l do s[i] = permrep(set,n,i) end for string mx = iff(hi=l?"":sprintf("/%d",l)), pof = sprintf("perms[%d..%d%s] of %v",{lo,hi,mx,set}) printf(1,"Len %d %-35s: %v\n",{n,pof,shorten(s[lo..hi],"",3)}) 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) show_all("XYZ",4,31,36) integer l = permrep("ACKR",5) for i=1 to l do if permrep("ACKR",5,i)="CRACK" then -- 455 printf(1,"Len 5 perm %d/%d of \"ACKR\" : 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))})
- Output:
Len 1 perms[1..3] of "123" : {"1","2","3"}
Len 2 perms[1..9] of "123" : {"11","12","13","...","31","32","33"}
Len 3 perms[1..27] of "123" : {"111","112","113","...","331","332","333"}
Len 3 perms[1..27] of "456" : {"444","445","446","...","664","665","666"}
Len 3 perms[1..27] of {1,2,3} : {{1,1,1},{1,1,2},{1,1,3},"...",{3,3,1},{3,3,2},{3,3,3}}
Len 2 perms[1..9] of {"bat","fox","cow"} : {{"bat","bat"},{"bat","fox"},{"bat","cow"},"...",{"cow","bat"},{"cow","fox"},{"cow","cow"}}
Len 4 perms[31..36/81] of "XYZ" : {"YXYX","YXYY","YXYZ","YXZX","YXZY","YXZZ"}
Len 5 perm 455/1024 of "ACKR" : CRACK
reverse(permrep("ACKR",5,589+1):CRACK
PHP
<?php
function permutate($values, $size, $offset) {
$count = count($values);
$array = array();
for ($i = 0; $i < $size; $i++) {
$selector = ($offset / pow($count,$i)) % $count;
$array[$i] = $values[$selector];
}
return $array;
}
function permutations($values, $size) {
$a = array();
$c = pow(count($values), $size);
for ($i = 0; $i<$c; $i++) {
$a[$i] = permutate($values, $size, $i);
}
return $a;
}
$permutations = permutations(['bat','fox','cow'], 2);
foreach ($permutations as $permutation) {
echo join(',', $permutation)."\n";
}
- Output:
bat,bat fox,bat cow,bat bat,fox fox,fox cow,fox bat,cow fox,cow cow,cow
PicoLisp
(de permrep (N Lst)
(if (=0 N)
(cons NIL)
(mapcan
'((X)
(mapcar '((Y) (cons Y X)) Lst) )
(permrep (dec N) Lst) ) ) )Python
Strict evaluation of the whole set
To evaluate the whole set of permutations, without the option to make complete evaluation conditional, we can reach for a generic replicateM function for lists:
'''Permutations of n elements drawn from k values'''
from itertools import product
# replicateM :: Applicative m => 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()
- Output:
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
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'):
'''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()
- Output:
Search for a 5 char permutation drawn from 'ACKR' matching "crack": Permutation 589 of 1024: CRACK
Quackery
A scenario for the task: An executive has forgotten the "combination" to unlock one of the clasps on their executive briefcase. It is 222 but they can't remember that. Unlikely as it may seem, they do remember that it does not have any zeros, or any numbers greater than 6. Also, the combination, when written as English words, "two two two" requires an odd number of letters. You'd think that, remembering details like that, they'd be able to recall the number itself, but such is the nature of programming tasks. <shrug>
Stepping through all the possibilities from 000 to 999 would take 3^10 steps, and is just a matter of counting from 0 to 999 inclusive, left padding the small numbers with zeros as required. As we know that some numbers are precluded we can reduce this to stepping from 000 to 444 in base 4, mapping the digits 0 to 4 onto the words "one" to "five", and printing only the resultant strings which have an odd number of characters.
Generators are not defined in Quackery, but are easy enough to create, requiring a single line of code.
[ ]this[ take ]'[ do ]this[ put ]done[ ] is generator ( --> )An explanation of how this works is beyond the scope of this task, but the use of "meta-words" (i.e. those wrapped in ]reverse-brackets[) is explored in The Book of Quackery. How generator can be used is illustrated in the somewhat trivial instance used in this task, counter, which returns 0 the first time is is called, and one more in every subsequent call. As a convenience we also define resetgen, which can be used to reset a generator word to a specified state.
[ ]'[ replace ] is resetgen ( x --> )As a microscopically less trivial example of words defined using generator and resetgen, the word fibonacci will return subsequent numbers on the Fibonacci sequence - 0, 1, 1, 2, 3, 5, 8… on each invocation, and can be restarted by calling resetfib.
[ generator [ do 2dup + join ] [ 0 1 ] ] is fibonacci ( --> n )
[ ' [ 0 1 ] resetgen fibonacci ] is resetfib ( --> )And so to the task:
[ 1 & ] is odd ( n --> b )
[ ]this[ take ]'[ do ]this[ put ]done[ ] is generator ( --> )
[ ]'[ replace ] is resetgen ( x --> )
[ generator [ dup 1+ ] 0 ] is counter ( --> n )
[ 0 resetgen counter ] is resetcounter ( --> n )
[ [] unrot times
[ base share /mod rot join swap ]
drop ] is ndigits ( n n --> [ )
[ [] unrot
over size base put
counter swap ndigits
witheach
[ dip dup peek
rot swap join
space join swap ]
drop
-1 split drop
base release ] is nextperm ( [ n --> [ )
[ [ $ "one two three four five"
nest$ ] constant
3 nextperm ] is task ( --> [ )
resetcounter
[ task
dup size odd if
[ dup echo$ cr ]
$ "two two two" = until ]- Output:
one one one one one two one one three one two one one two two one two three one three one one three two one three three one four four one four five one five four one five five two one one two one two two one three two two one two two two
Racket
As a sequence
First we define a procedure that defines the sequence of the permutations.
#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)))
- Output:
'((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.
(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)))
- Output:
'((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))
Raku
(formerly Perl 6)
We can use the X operator ("cartesian product") to cross the list with itself.
For :
my @k = <a b c>;
.say for @k X @k;
For arbitrary :
my @k = <a b c>;
my $n = 2;
.say for [X] @k xx $n;
- Output:
a a a b a c b a b b b c c a c b c c
Here is an other approach, counting all possibilities in base :
my @k = <a b c>;
my $n = 2;
say @k[.polymod: +@k xx $n-1] for ^@k**$n
- Output:
a a b a c a a b b b c b a c b c c c
REXX
version 1
/*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.*/
- output when using the default inputs of: 3 2
11 12 13 21 22 23 31 32 33
- output when using the default inputs of : 3 2 , bat fox cow
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 ***************************************************************
* 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 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. */
- output when using the following inputs: 3 2
11 12 13 21 22 23 31 32 33
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 nlOutput:
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):
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]]
Rust
struct PermutationIterator<'a, T: 'a> {
universe: &'a [T],
size: usize,
prev: Option<Vec<usize>>,
}
fn permutations<T>(universe: &[T], size: usize) -> PermutationIterator<T> {
PermutationIterator {
universe,
size,
prev: None,
}
}
fn map<T>(values: &[T], ixs: &[usize]) -> Vec<T>
where
T: Clone,
{
ixs.iter().map(|&i| values[i].clone()).collect()
}
impl<'a, T> Iterator for PermutationIterator<'a, T>
where
T: Clone,
{
type Item = Vec<T>;
fn next(&mut self) -> Option<Vec<T>> {
let n = self.universe.len();
if n == 0 {
return None;
}
match self.prev {
None => {
let zeroes: Vec<usize> = std::iter::repeat(0).take(self.size).collect();
let result = Some(map(self.universe, &zeroes[..]));
self.prev = Some(zeroes);
result
}
Some(ref mut indexes) => match indexes.iter().position(|&i| i + 1 < n) {
None => None,
Some(position) => {
for index in indexes.iter_mut().take(position) {
*index = 0;
}
indexes[position] += 1;
Some(map(self.universe, &indexes[..]))
}
},
}
}
}
fn main() {
let universe = ["Annie", "Barbie"];
for p in permutations(&universe[..], 3) {
for element in &p {
print!("{} ", element);
}
println!();
}
}
- Output:
Annie Annie Annie Barbie Annie Annie Annie Barbie Annie Barbie Barbie Annie Annie Annie Barbie Barbie Annie Barbie Annie Barbie Barbie Barbie Barbie Barbie
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))
}
- Output:
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
var k = %w(a b c)
var n = 2
cartesian([k] * n, {|*a| say a.join(' ') })
- Output:
a a a b a c b a b b b c c a c b c c
Standard ML
fun multiperms [] _ = [[]]
| multiperms _ 0 = [[]]
| multiperms xs n =
let
val rest = multiperms xs (n-1)
in
List.concat (List.map (fn a => (List.map (fn b => a::b) rest)) xs)
end
Tcl
Iterative version
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
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
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
}
Wren
var n = 3
var values = ["A", "B", "C", "D"]
var k = values.count
// terminate when first two characters of the permutation are 'B' and 'C' respectively
var decide = Fn.new { |pc| pc[0] == "B" && pc[1] == "C" }
var pn = List.filled(n, 0)
var pc = List.filled(n, null)
while (true) {
// generate permutation
var i = 0
for (x in pn) {
pc[i] = values[x]
i = i + 1
}
// show progress
System.print(pc)
// pass to deciding function
if (decide.call(pc)) return // terminate early
// increment permutation number
i = 0
while (true) {
pn[i] = pn[i] + 1
if (pn[i] < k) break
pn[i] = 0
i = i + 1
if (i == n) return // all permutations generated
}
}
- Output:
[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]
XPL0
func Decide(PC);
\Terminate when first two characters of permutation are 'B' and 'C' respectively
int PC;
return PC(0)=^B & PC(1)=^C;
def N=3, K=4;
int Values, PN(N), PC(N), I, X;
[Values:= [^A, ^B, ^C, ^D];
for I:= 0 to N-1 do PN(I):= 0;
loop [for I:= 0 to N-1 do
[X:= PN(I);
PC(I):= Values(X);
];
ChOut(0, ^[); \show progress
for I:= 0 to N-1 do
[if I # 0 then Text(0, ", "); ChOut(0, PC(I))];
ChOut(0, ^]); CrLf(0);
\pass to deciding function
if Decide(PC) then return; \terminate early
I:= 0; \increment permutation number
loop [PN(I):= PN(I)+1;
if PN(I) < K then quit;
PN(I):= 0;
I:= I+1;
if I = N then return; \all permutations generated
];
];
]- Output:
[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]
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