Permutations by swapping

From Rosetta Code
Task
Permutations by swapping
You are encouraged to solve this task according to the task description, using any language you may know.
Task

Generate permutations of n items in which successive permutations differ from each other by the swapping of any two items.

Also generate the sign of the permutation which is +1 when the permutation is generated from an even number of swaps from the initial state, and -1 for odd.

Show the permutations and signs of three items, in order of generation here.

Such data are of use in generating the determinant of a square matrix and any functions created should bear this in mind.

Note: The Steinhaus–Johnson–Trotter algorithm generates successive permutations where adjacent items are swapped, but from this discussion adjacency is not a requirement.


References


Related tasks



11l

F s_permutations(seq)
   V items = [[Int]()]
   L(j) seq
      [[Int]] new_items
      L(item) items
         I L.index % 2
            new_items [+]= (0..item.len).map(i -> @item[0 .< i] [+] [@j] [+] @item[i..])
         E
            new_items [+]= (item.len..0).step(-1).map(i -> @item[0 .< i] [+] [@j] [+] @item[i..])
      items = new_items

   R enumerate(items).map((i, item) -> (item, I i % 2 {-1} E 1))

L(n) (3, 4)
   print(‘Permutations and sign of #. items’.format(n))
   L(perm, sgn) s_permutations(Array(0 .< n))
      print(‘Perm: #. Sign: #2’.format(perm, sgn))
   print()
Output:
Permutations and sign of 3 items
Perm: [0, 1, 2] Sign:  1
Perm: [0, 2, 1] Sign: -1
Perm: [2, 0, 1] Sign:  1
Perm: [2, 1, 0] Sign: -1
Perm: [1, 2, 0] Sign:  1
Perm: [1, 0, 2] Sign: -1

Permutations and sign of 4 items
Perm: [0, 1, 2, 3] Sign:  1
Perm: [0, 1, 3, 2] Sign: -1
Perm: [0, 3, 1, 2] Sign:  1
Perm: [3, 0, 1, 2] Sign: -1
Perm: [3, 0, 2, 1] Sign:  1
Perm: [0, 3, 2, 1] Sign: -1
Perm: [0, 2, 3, 1] Sign:  1
Perm: [0, 2, 1, 3] Sign: -1
Perm: [2, 0, 1, 3] Sign:  1
Perm: [2, 0, 3, 1] Sign: -1
Perm: [2, 3, 0, 1] Sign:  1
Perm: [3, 2, 0, 1] Sign: -1
Perm: [3, 2, 1, 0] Sign:  1
Perm: [2, 3, 1, 0] Sign: -1
Perm: [2, 1, 3, 0] Sign:  1
Perm: [2, 1, 0, 3] Sign: -1
Perm: [1, 2, 0, 3] Sign:  1
Perm: [1, 2, 3, 0] Sign: -1
Perm: [1, 3, 2, 0] Sign:  1
Perm: [3, 1, 2, 0] Sign: -1
Perm: [3, 1, 0, 2] Sign:  1
Perm: [1, 3, 0, 2] Sign: -1
Perm: [1, 0, 3, 2] Sign:  1
Perm: [1, 0, 2, 3] Sign: -1

Ada

-- Permutations by swapping
-- J. Carter     2024 Jun
-- Uses the PragmAda Reusable Components (https://github.com/jrcarter/PragmARC)

with Ada.Text_IO;
with PragmARC.Permutations;

procedure Permutations_By_Swapping is
   package Permutations is new PragmARC.Permutations (Element => Positive);
   
   Permutation : Permutations.Sequence_Lists.Vector;
begin -- Permutations_By_Swapping
   Permutations.Generate (Initial => (1, 2, 3), Result => Permutation);
   
   Output : for I in 1 .. Permutation.Last_Index loop
      One_Permutation : declare
         Sequence : Permutations.Sequence renames Permutation.Element (I);
      begin -- One_Permutation
         Ada.Text_IO.Put (Item => '(');
         
         Print : for J in Sequence'Range loop
            Ada.Text_IO.Put (Item => (if J > 1 then "," else "") & Sequence (J)'Image);
         end loop Print;
         
         Ada.Text_IO.Put_Line (Item => ") " & (if I rem 2 = 0 then '-' else '+') & '1');
      end One_Permutation;
   end loop Output;
end Permutations_By_Swapping;
Output:
( 1, 2, 3) +1
( 2, 1, 3) -1
( 3, 1, 2) +1
( 1, 3, 2) -1
( 2, 3, 1) +1
( 3, 2, 1) -1

ALGOL 68

Based on the pseudo-code for the recursive version of Heap's algorithm.

BEGIN # Heap's algorithm for generating permutations - from the pseudo-code on the Wikipedia page #
    # generate permutations of a #
    PROC generate = ( INT k, REF[]INT a, REF INT swap count )VOID:
         IF k = 1 THEN
            output permutation( a, swap count )
         ELSE
            # Generate permutations with kth unaltered #
            # Initially k = length a #
            generate( k - 1, a, swap count );
            # Generate permutations for kth swapped with each k-1 initial #
            FOR i FROM 0 TO k - 2 DO
                # Swap choice dependent on parity of k (even or odd) #
                swap count +:= 1;
                INT swap item = IF ODD k THEN 0 ELSE i FI;
                INT t           = a[ swap item ];
                a[ swap item ] := a[ k - 1 ];
                a[ k - 1     ] := t;
                generate( k - 1, a, swap count )
            OD
         FI # generate # ;
    # generate permutations of a #
    PROC permute = ( REF[]INT a )VOID:
         BEGIN
            INT swap count := 0;
            generate( ( UPB a + 1 ) - LWB a, a[ AT 0 ], swap count )
         END # permute # ;
 
    # handle a permutation #
    PROC output permutation = ( REF[]INT a, INT swap count )VOID:
         BEGIN
            print( ( "[" ) );
            FOR i FROM LWB a TO UPB a DO
               print( ( whole( a[ i ], 0 ) ) );
               IF i = UPB a THEN print( ( "]" ) ) ELSE print( ( ", " ) ) FI
            OD;
            print( ( " sign: ", IF ODD swap count THEN "-1" ELSE " 1" FI, newline ) )
         END # output permutation # ;

    [ 1 : 3 ]INT a := ( 1, 2, 3 );
    permute( a )

END
Output:
[1, 2, 3] sign:  1
[2, 1, 3] sign: -1
[3, 1, 2] sign:  1
[1, 3, 2] sign: -1
[2, 3, 1] sign:  1
[3, 2, 1] sign: -1

Arturo

permutations: function [arr][
    d: 1
    c: array.of: size arr 0
    xs: new arr
    sign: 1

    ret: new @[@[xs, sign]]

    while [true][
        while [d > 1][
            d: d-1
            c\[d]: 0
        ]

        while [c\[d] >= d][
            d: d+1
            if d >= size arr -> return ret
        ]

        i: (1 = and d 1)? -> c\[d] -> 0
        tmp: xs\[i]
        xs\[i]: xs\[d]
        xs\[d]: tmp

        sign: neg sign
        'ret ++ @[new @[xs, sign]]
        c\[d]: c\[d] + 1
    ]

    return ret
]

loop permutations 0..2 'row ->
    print [row\0 "-> sign:" row\1]

print ""

loop permutations 0..3 'row ->
    print [row\0 "-> sign:" row\1]
Output:
[0 1 2] -> sign: 1 
[1 0 2] -> sign: -1 
[2 0 1] -> sign: 1 
[0 2 1] -> sign: -1 
[1 2 0] -> sign: 1 
[2 1 0] -> sign: -1 

[0 1 2 3] -> sign: 1 
[1 0 2 3] -> sign: -1 
[2 0 1 3] -> sign: 1 
[0 2 1 3] -> sign: -1 
[1 2 0 3] -> sign: 1 
[2 1 0 3] -> sign: -1 
[3 1 0 2] -> sign: 1 
[1 3 0 2] -> sign: -1 
[0 3 1 2] -> sign: 1 
[3 0 1 2] -> sign: -1 
[1 0 3 2] -> sign: 1 
[0 1 3 2] -> sign: -1 
[0 2 3 1] -> sign: 1 
[2 0 3 1] -> sign: -1 
[3 0 2 1] -> sign: 1 
[0 3 2 1] -> sign: -1 
[2 3 0 1] -> sign: 1 
[3 2 0 1] -> sign: -1 
[3 2 1 0] -> sign: 1 
[2 3 1 0] -> sign: -1 
[1 3 2 0] -> sign: 1 
[3 1 2 0] -> sign: -1 
[2 1 3 0] -> sign: 1 
[1 2 3 0] -> sign: -1

AutoHotkey

Permutations_By_Swapping(str, list:=""){
	ch := SubStr(str, 1, 1)								; get left-most charachter of str
	for i, line in StrSplit(list, "`n")					; for each line in list
		loop % StrLen(line) + 1							; loop each possible position
			Newlist .= RegExReplace(line, mod(i,2) ? "(?=.{" A_Index-1 "}$)" : "^.{" A_Index-1 "}\K", ch) "`n"
	list := Newlist ? Trim(Newlist, "`n") : ch			; recreate list
	if !str := SubStr(str, 2)							; remove charachter from left hand side
		return list										; done if str is empty
	return Permutations_By_Swapping(str, list)			; else recurse
}

Examples:

for each, line in StrSplit(Permutations_By_Swapping(1234), "`n")
	result .= line "`tSign: " (mod(A_Index,2)? 1 : -1) "`n"
MsgBox, 262144, , % result
return

Outputs:

1234	Sign: 1
1243	Sign: -1
1423	Sign: 1
4123	Sign: -1
4132	Sign: 1
1432	Sign: -1
1342	Sign: 1
1324	Sign: -1
3124	Sign: 1
3142	Sign: -1
3412	Sign: 1
4312	Sign: -1
4321	Sign: 1
3421	Sign: -1
3241	Sign: 1
3214	Sign: -1
2314	Sign: 1
2341	Sign: -1
2431	Sign: 1
4231	Sign: -1
4213	Sign: 1
2413	Sign: -1
2143	Sign: 1
2134	Sign: -1

BASIC

BASIC256

Translation of: Free BASIC
call perms(3)
print
call perms(4)
end

subroutine perms(n)
    dim p((n+1)*4)
    for i = 1 to n
        p[i] = -i
    next i
    s = 1
    
    do
        print "Perm: [ ";
        for i = 1 to n
            print abs(p[i]); " ";
        next i
        print "] Sign: "; s

        k = 0
        for i = 2 to n
            if p[i] < 0 and (abs(p[i]) > abs(p[i-1])) and (abs(p[i]) > abs(p[k])) then k = i
        next i
        for i = 1 to n-1
            if p[i] > 0 and (abs(p[i]) > abs(p[i+1])) and (abs(p[i]) > abs(p[k])) then k = i
        next i
        if k then
            for i = 1 to n    #reverse elements > k
                if abs(p[i]) > abs(p[k]) then p[i] = -p[i]
            next i
            if p[k] < 0 then i = k-1 else i = k+1
            temp = p[k]
            p[k] = p[i]
            p[i] = temp
            s = -s
        end if
    until k = 0
end subroutine

QBasic

Works with: QBasic version 1.1
Works with: QuickBasic version 4.5
Translation of: Free BASIC
SUB perms (n)
    DIM p((n + 1) * 4)
    FOR i = 1 TO n
    p(i) = -i
    NEXT i
    s = 1
    
    DO
    PRINT "Perm: (";
    FOR i = 1 TO n
        PRINT ABS(p(i)); "";
    NEXT i
    PRINT ") Sign: "; s

    k = 0
    FOR i = 2 TO n
        IF p(i) < 0 AND (ABS(p(i)) > ABS(p(i - 1))) AND (ABS(p(i)) > ABS(p(k))) THEN k = i
    NEXT i
    FOR i = 1 TO n - 1
        IF p(i) > 0 AND (ABS(p(i)) > ABS(p(i + 1))) AND (ABS(p(i)) > ABS(p(k))) THEN k = i
    NEXT i
    IF k THEN
        FOR i = 1 TO n    'reverse elements > k
        IF ABS(p(i)) > ABS(p(k)) THEN p(i) = -p(i)
        NEXT i
        'if p(k) < 0 then i = k-1 else i = k+1
        i = k + SGN(p(k))
        SWAP p(k), p(i)
        'temp = p(k)
        'p(k) = p(i)
        'p(i) = temp
        s = -s
    END IF
    LOOP UNTIL k = 0
END SUB

perms (3)
PRINT
perms (4)

Run BASIC

Translation of: Free BASIC
sub perms n
    dim p((n+1)*4)
    for i = 1 to n : p(i) = i*-1 : next i
    s = 1
    
    while 1
        print "Perm: [ ";
        for i = 1 to n
            print abs(p(i)); " ";
        next i
        print "] Sign: "; s

        k = 0
        for i = 2 to n
            if p(i) < 0 and (abs(p(i)) > abs(p(i-1))) and (abs(p(i)) > abs(p(k))) then k = i
        next i
        for i = 1 to n-1
            if p(i) > 0 and (abs(p(i)) > abs(p(i+1))) and (abs(p(i)) > abs(p(k))) then k = i
        next i
        if k then
            for i = 1 to n   'reverse elements > k
                if abs(p(i)) > abs(p(k)) then p(i) = p(i)*-1
            next i
            if p(k) < 0 then i = k-1 else i = k+1   'swap K with element looked at
            temp = p(k)
            p(k) = p(i)
            p(i) = temp
            s = s*-1         'alternate signs
        end if
    if k = 0 then exit while
    wend
end sub

call perms 3
print
call perms 4

Yabasic

Translation of: Free BASIC
perms(3)
print
perms(4)
end

sub perms(n)
    dim p((n+1)*4)
    for i = 1 to n
        p(i) = -i
    next i
    s = 1
    
    repeat
        print "Perm: [ ";
        for i = 1 to n
            print abs(p(i)), " ";
        next i
        print "] Sign: ", s

        k = 0
        for i = 2 to n
            if p(i) < 0 and (abs(p(i)) > abs(p(i-1))) and (abs(p(i)) > abs(p(k)))  k = i
        next i
        for i = 1 to n-1
            if p(i) > 0 and (abs(p(i)) > abs(p(i+1))) and (abs(p(i)) > abs(p(k)))  k = i
        next i
        if k then
            for i = 1 to n    //reverse elements > k
                if abs(p(i)) > abs(p(k))  p(i) = -p(i)
            next i
            i = k + sig(p(k))
            temp = p(k)
            p(k) = p(i)
            p(i) = temp
            s = -s
        endif
    until k = 0
end sub

BBC BASIC

      PROCperms(3)
      PRINT
      PROCperms(4)
      END
      
      DEF PROCperms(n%)
      LOCAL p%(), i%, k%, s%
      DIM p%(n%)
      FOR i% = 1 TO n%
        p%(i%) = -i%
      NEXT
      s% = 1
      REPEAT
        PRINT "Perm: [ ";
        FOR i% = 1 TO n%
          PRINT ;ABSp%(i%) " ";
        NEXT
        PRINT "] Sign: ";s%
        k% = 0
        FOR i% = 2 TO n%
          IF p%(i%)<0 IF ABSp%(i%)>ABSp%(i%-1) IF ABSp%(i%)>ABSp%(k%) k% = i%
        NEXT
        FOR i% = 1 TO n%-1
          IF p%(i%)>0 IF ABSp%(i%)>ABSp%(i%+1) IF ABSp%(i%)>ABSp%(k%) k% = i%
        NEXT
        IF k% THEN
          FOR i% = 1 TO n%
            IF ABSp%(i%)>ABSp%(k%) p%(i%) *= -1
          NEXT
          i% = k%+SGNp%(k%)
          SWAP p%(k%),p%(i%)
          s% = -s%
        ENDIF
      UNTIL k% = 0
      ENDPROC
Output:
Perm: [ 1 2 3 ] Sign: 1
Perm: [ 1 3 2 ] Sign: -1
Perm: [ 3 1 2 ] Sign: 1
Perm: [ 3 2 1 ] Sign: -1
Perm: [ 2 3 1 ] Sign: 1
Perm: [ 2 1 3 ] Sign: -1

Perm: [ 1 2 3 4 ] Sign: 1
Perm: [ 1 2 4 3 ] Sign: -1
Perm: [ 1 4 2 3 ] Sign: 1
Perm: [ 4 1 2 3 ] Sign: -1
Perm: [ 4 1 3 2 ] Sign: 1
Perm: [ 1 4 3 2 ] Sign: -1
Perm: [ 1 3 4 2 ] Sign: 1
Perm: [ 1 3 2 4 ] Sign: -1
Perm: [ 3 1 2 4 ] Sign: 1
Perm: [ 3 1 4 2 ] Sign: -1
Perm: [ 3 4 1 2 ] Sign: 1
Perm: [ 4 3 1 2 ] Sign: -1
Perm: [ 4 3 2 1 ] Sign: 1
Perm: [ 3 4 2 1 ] Sign: -1
Perm: [ 3 2 4 1 ] Sign: 1
Perm: [ 3 2 1 4 ] Sign: -1
Perm: [ 2 3 1 4 ] Sign: 1
Perm: [ 2 3 4 1 ] Sign: -1
Perm: [ 2 4 3 1 ] Sign: 1
Perm: [ 4 2 3 1 ] Sign: -1
Perm: [ 4 2 1 3 ] Sign: 1
Perm: [ 2 4 1 3 ] Sign: -1
Perm: [ 2 1 4 3 ] Sign: 1
Perm: [ 2 1 3 4 ] Sign: -1

C

Implementation of Heap's Algorithm, array length has to be passed as a parameter for non character arrays, as sizeof() will not give correct results when malloc is used. Prints usage on incorrect invocation.

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

int flag = 1;

void heapPermute(int n, int arr[],int arrLen){
	int temp;
	int i;
	
	if(n==1){
		printf("\n[");
		
		for(i=0;i<arrLen;i++)
			printf("%d,",arr[i]);
		printf("\b] Sign : %d",flag);
		
		flag*=-1;
	}
	else{
		for(i=0;i<n-1;i++){
			heapPermute(n-1,arr,arrLen);
			
			if(n%2==0){
				temp = arr[i];
				arr[i] = arr[n-1];
				arr[n-1] = temp;
			}
			else{
				temp = arr[0];
				arr[0] = arr[n-1];
				arr[n-1] = temp;
			}
		}
		heapPermute(n-1,arr,arrLen);
	}
}

int main(int argC,char* argV[0])
{
	int *arr, i=0, count = 1;
	char* token;
	
	if(argC==1)
		printf("Usage : %s <comma separated list of integers>",argV[0]);
	else{
		while(argV[1][i]!=00){
			if(argV[1][i++]==',')
				count++;
		}
		
		arr = (int*)malloc(count*sizeof(int));
		
		i = 0;
		
		token = strtok(argV[1],",");
		
		while(token!=NULL){
			arr[i++] = atoi(token);
			token = strtok(NULL,",");
		}
		
		heapPermute(i,arr,count);
	}
		
	return 0;
}

Output:

C:\rosettaCode>heapPermute.exe 1,2,3

[1,2,3] Sign : 1
[2,1,3] Sign : -1
[3,1,2] Sign : 1
[1,3,2] Sign : -1
[2,3,1] Sign : 1
[3,2,1] Sign : -1

C++

Direct implementation of Johnson-Trotter algorithm from the reference link.

#include <iostream>
#include <vector>

using namespace std;

vector<int> UpTo(int n, int offset = 0)
{
	vector<int> retval(n);
	for (int ii = 0; ii < n; ++ii)
		retval[ii] = ii + offset;
	return retval;
}

struct JohnsonTrotterState_
{
	vector<int> values_;
	vector<int> positions_;	// size is n+1, first element is not used
	vector<bool> directions_;
	int sign_;

	JohnsonTrotterState_(int n) : values_(UpTo(n, 1)), positions_(UpTo(n + 1, -1)), directions_(n + 1, false), sign_(1) {}

	int LargestMobile() const	// returns 0 if no mobile integer exists
	{
		for (int r = values_.size(); r > 0; --r)
		{
			const int loc = positions_[r] + (directions_[r] ? 1 : -1);
			if (loc >= 0 && loc < values_.size() && values_[loc] < r)
				return r;
		}
		return 0;
	}

	bool IsComplete() const { return LargestMobile() == 0; }

	void operator++()	// implement Johnson-Trotter algorithm
	{
		const int r = LargestMobile();
		const int rLoc = positions_[r];
		const int lLoc = rLoc + (directions_[r] ? 1 : -1);
		const int l = values_[lLoc];
		// do the swap
		swap(values_[lLoc], values_[rLoc]);
		swap(positions_[l], positions_[r]);
		sign_ = -sign_;
		// change directions
		for (auto pd = directions_.begin() + r + 1; pd != directions_.end(); ++pd)
			*pd = !*pd;
	}
};

int main(void)
{
	JohnsonTrotterState_ state(4);
	do
	{
		for (auto v : state.values_)
			cout << v << " ";
		cout << "\n";
		++state;
	} while (!state.IsComplete());
}
Output:
(1 2 3 4 ); sign = 1
(1 2 4 3 ); sign = -1
(1 4 2 3 ); sign = 1
(4 1 2 3 ); sign = -1
(4 1 3 2 ); sign = 1
(1 4 3 2 ); sign = -1
(1 3 4 2 ); sign = 1
(1 3 2 4 ); sign = -1
(3 1 2 4 ); sign = 1
(3 1 4 2 ); sign = -1
(3 4 1 2 ); sign = 1
(4 3 1 2 ); sign = -1
(4 3 2 1 ); sign = 1
(3 4 2 1 ); sign = -1
(3 2 4 1 ); sign = 1
(3 2 1 4 ); sign = -1
(2 3 1 4 ); sign = 1
(2 3 4 1 ); sign = -1
(2 4 3 1 ); sign = 1
(4 2 3 1 ); sign = -1
(4 2 1 3 ); sign = 1
(2 4 1 3 ); sign = -1
(2 1 4 3 ); sign = 1

Clojure

Recursive version

(defn permutation-swaps
  "List of swap indexes to generate all permutations of n elements"
  [n]
  (if (= n 2) `((0 1))
    (let [old-swaps (permutation-swaps (dec n))
          swaps-> (partition 2 1 (range n))
          swaps<- (reverse swaps->)]
      (mapcat (fn [old-swap side]
                (case side
                  :first swaps<-
                  :right (conj swaps<- old-swap)
                  :left (conj swaps-> (map inc old-swap))))
              (conj old-swaps nil)
              (cons :first (cycle '(:left :right)))))))


(defn swap [v [i j]]
  (-> v
      (assoc i (nth v j))
      (assoc j (nth v i))))


(defn permutations [n]
  (let [permutations (reduce
                       (fn [all-perms new-swap]
                         (conj all-perms (swap (last all-perms)
                                               new-swap)))
                       (vector (vec (range n)))
                       (permutation-swaps n))
        output (map vector
                    permutations
                    (cycle '(1 -1)))]
    output))


(doseq [n [2 3 4]]
  (dorun (map println (permutations n))))
Output:
[[0 1] 1]
[[1 0] -1]
[[0 1 2] 1]
[[0 2 1] -1]
[[2 0 1] 1]
[[2 1 0] -1]
[[1 2 0] 1]
[[1 0 2] -1]
[[0 1 2 3] 1]
[[0 1 3 2] -1]
[[0 3 1 2] 1]
[[3 0 1 2] -1]
[[3 0 2 1] 1]
[[0 3 2 1] -1]
[[0 2 3 1] 1]
[[0 2 1 3] -1]
[[2 0 1 3] 1]
[[2 0 3 1] -1]
[[2 3 0 1] 1]
[[3 2 0 1] -1]
[[3 2 1 0] 1]
[[2 3 1 0] -1]
[[2 1 3 0] 1]
[[2 1 0 3] -1]
[[1 2 0 3] 1]
[[1 2 3 0] -1]
[[1 3 2 0] 1]
[[3 1 2 0] -1]
[[3 1 0 2] 1]
[[1 3 0 2] -1]
[[1 0 3 2] 1]
[[1 0 2 3] -1]

Modeled After Python version

Translation of: Python
(ns test-p.core)

(defn numbers-only [x]
  " Just shows the numbers only for the pairs (i.e. drops the direction --used for display purposes when printing the result"
  (mapv first x))

(defn next-permutation
  " Generates next permutation from the current (p) using the Johnson-Trotter technique
    The code below translates the Python version which has the following steps:
     p of form [...[n dir]...] such as [[0 1] [1 1] [2 -1]], where n is a number and dir = direction (=1=right, -1=left, 0=don't move)
     Step: 1 finds the pair [n dir] with the largest value of n (where dir is not equal to 0 (done if none)
     Step: 2: swap the max pair found with its neighbor in the direction of the pair (i.e. +1 means swap to right, -1 means swap left
     Step 3: if swapping places the pair a the beginning or end of the list, set the direction = 0 (i.e. becomes non-mobile)
     Step 4: Set the directions of all pairs whose numbers are greater to the right of where the pair was moved to -1 and to the left to +1 "
  [p]
  (if (every? zero? (map second p))
    nil                                                                 ; no mobile elements (all directions are zero)
    (let [n (count p)
          ; Step 1
          fn-find-max (fn [m]
                        (first (apply max-key                           ; find the max mobile elment
                                   (fn [[i x]]
                                     (if (zero? (second x))
                                       -1
                                       (first x)))
                                              (map-indexed vector p))))
          i1 (fn-find-max p)                                            ; index of max
          [n1 d1] (p i1)                                                ; value and direction of max
          i2 (+ d1 i1)
          fn-swap (fn [m] (assoc m i2 (m i1) i1 (m i2)))                ; function to swap with neighbor in our step direction
          fn-update-max (fn [m] (if (or (contains? #{0 (dec n)} i2)     ; update direction of max (where max went)
                                        (> ((m (+ i2 d1)) 0) n1))
                                  (assoc-in m [i2 1] 0)
                                  m))
          fn-update-others (fn [[i3 [n3 d3]]]                            ; Updates directions of pairs to the left and right of max
                             (cond                                       ; direction reset to -1 if to right, +1 if to left
                               (<= n3 n1) [n3 d3]
                               (< i3 i2) [n3 1]
                               :else      [n3 -1]))]
      ; apply steps 2, 3, 4(using functions that where created for these steps)
      (mapv fn-update-others (map-indexed vector (fn-update-max (fn-swap p)))))))

(defn spermutations
  " Lazy sequence of permutations of n digits"
  ; Each element is two element vector (number direction)
  ; Startup case - generates sequence 0...(n-1) with move direction (1 = move right, -1 = move left, 0 = don't move)
  ([n] (spermutations 1
                      (into [] (for [i (range n)] (if (zero? i)
                                                    [i 0]               ; 0th element is not mobile yet
                                                    [i -1])))))         ; all others move left
  ([sign p]
   (when-let [s (seq p)]
             (cons [(numbers-only p) sign]
                   (spermutations (- sign) (next-permutation p))))))   ; recursively tag onto sequence


;; Print results for 2, 3, and 4 items
(doseq [n (range 2 5)]
  (do
    (println)
    (println (format "Permutations and sign of %d items " n))
  (doseq [q (spermutations n)] (println (format "Perm: %s Sign: %2d" (first q) (second q))))))
Output:
Permutations and sign of 2 items 
Perm: [0 1] Sign:  1
Perm: [1 0] Sign: -1

Permutations and sign of 3 items 
Perm: [0 1 2] Sign:  1
Perm: [0 2 1] Sign: -1
Perm: [2 0 1] Sign:  1
Perm: [2 1 0] Sign: -1
Perm: [1 2 0] Sign:  1
Perm: [1 0 2] Sign: -1

Permutations and sign of 4 items 
Perm: [0 1 2 3] Sign:  1
Perm: [0 1 3 2] Sign: -1
Perm: [0 3 1 2] Sign:  1
Perm: [3 0 1 2] Sign: -1
Perm: [3 0 2 1] Sign:  1
Perm: [0 3 2 1] Sign: -1
Perm: [0 2 3 1] Sign:  1
Perm: [0 2 1 3] Sign: -1
Perm: [2 0 1 3] Sign:  1
Perm: [2 0 3 1] Sign: -1
Perm: [2 3 0 1] Sign:  1
Perm: [3 2 0 1] Sign: -1
Perm: [3 2 1 0] Sign:  1
Perm: [2 3 1 0] Sign: -1
Perm: [2 1 3 0] Sign:  1
Perm: [2 1 0 3] Sign: -1
Perm: [1 2 0 3] Sign:  1
Perm: [1 2 3 0] Sign: -1
Perm: [1 3 2 0] Sign:  1
Perm: [3 1 2 0] Sign: -1
Perm: [3 1 0 2] Sign:  1
Perm: [1 3 0 2] Sign: -1
Perm: [1 0 3 2] Sign:  1
Perm: [1 0 2 3] Sign: -1

Common Lisp

(defstruct (directed-number (:conc-name dn-))
  (number nil :type integer)
  (direction nil :type (member :left :right)))

(defmethod print-object ((dn directed-number) stream)
  (ecase (dn-direction dn)
    (:left  (format stream "<~D" (dn-number dn)))
    (:right (format stream "~D>" (dn-number dn)))))

(defun dn> (dn1 dn2)
  (declare (directed-number dn1 dn2))
  (> (dn-number dn1) (dn-number dn2)))

(defun dn-reverse-direction (dn)
  (declare (directed-number dn))
  (setf (dn-direction dn) (ecase (dn-direction dn)
                            (:left  :right)
                            (:right :left))))

(defun make-directed-numbers-upto (upto)
  (let ((numbers (make-array upto :element-type 'integer)))
    (dotimes (n upto numbers)
      (setf (aref numbers n) (make-directed-number :number (1+ n) :direction :left)))))

(defun max-mobile-pos (numbers)
  (declare ((vector directed-number) numbers))
  (loop with pos-limit = (1- (length numbers))
        with max-value and max-pos
        for num across numbers
        for pos from 0
        do (ecase (dn-direction num)
             (:left  (when (and (plusp pos) (dn> num (aref numbers (1- pos)))
                                (or (null max-value) (dn> num max-value)))
                       (setf max-value num
                             max-pos   pos)))
             (:right (when (and (< pos pos-limit) (dn> num (aref numbers (1+ pos)))
                                (or (null max-value) (dn> num max-value)))
                       (setf max-value num
                             max-pos   pos))))
        finally (return max-pos)))

(defun permutations (upto)
  (loop with numbers = (make-directed-numbers-upto upto)
        for max-mobile-pos = (max-mobile-pos numbers)
        for sign = 1 then (- sign)
        do (format t "~A sign: ~:[~;+~]~D~%" numbers (plusp sign) sign)
        while max-mobile-pos
        do (let ((max-mobile-number (aref numbers max-mobile-pos)))
             (ecase (dn-direction max-mobile-number)
               (:left  (rotatef (aref numbers (1- max-mobile-pos))
                                (aref numbers max-mobile-pos)))
               (:right (rotatef (aref numbers max-mobile-pos)
                                (aref numbers (1+ max-mobile-pos)))))
             (loop for n across numbers
                   when (dn> n max-mobile-number)
                     do (dn-reverse-direction n)))))

(permutations 3)
(permutations 4)
Output:
#(<1 <2 <3) sign: +1
#(<1 <3 <2) sign: -1
#(<3 <1 <2) sign: +1
#(3> <2 <1) sign: -1
#(<2 3> <1) sign: +1
#(<2 <1 3>) sign: -1
#(<1 <2 <3 <4) sign: +1
#(<1 <2 <4 <3) sign: -1
#(<1 <4 <2 <3) sign: +1
#(<4 <1 <2 <3) sign: -1
#(4> <1 <3 <2) sign: +1
#(<1 4> <3 <2) sign: -1
#(<1 <3 4> <2) sign: +1
#(<1 <3 <2 4>) sign: -1
#(<3 <1 <2 <4) sign: +1
#(<3 <1 <4 <2) sign: -1
#(<3 <4 <1 <2) sign: +1
#(<4 <3 <1 <2) sign: -1
#(4> 3> <2 <1) sign: +1
#(3> 4> <2 <1) sign: -1
#(3> <2 4> <1) sign: +1
#(3> <2 <1 4>) sign: -1
#(<2 3> <1 <4) sign: +1
#(<2 3> <4 <1) sign: -1
#(<2 <4 3> <1) sign: +1
#(<4 <2 3> <1) sign: -1
#(4> <2 <1 3>) sign: +1
#(<2 4> <1 3>) sign: -1
#(<2 <1 4> 3>) sign: +1
#(<2 <1 3> 4>) sign: -1

D

Iterative Version

This isn't a Range yet.

Translation of: Python
import std.algorithm, std.array, std.typecons, std.range;

struct Spermutations(bool doCopy=true) {
    private immutable uint n;
    alias TResult = Tuple!(int[], int);

    int opApply(in int delegate(in ref TResult) nothrow dg) nothrow {
        int result;

        int sign = 1;
        alias Int2 = Tuple!(int, int);
        auto p = n.iota.map!(i => Int2(i, i ? -1 : 0)).array;
        TResult aux;

        aux[0] = p.map!(pi => pi[0]).array;
        aux[1] = sign;
        result = dg(aux);
        if (result)
            goto END;

        while (p.any!q{ a[1] }) {
            // Failed to use std.algorithm here, too much complex.
            auto largest = Int2(-100, -100);
            int i1 = -1;
            foreach (immutable i, immutable pi; p)
                if (pi[1])
                    if (pi[0] > largest[0]) {
                        i1 = i;
                        largest = pi;
                    }
            immutable n1 = largest[0],
                      d1 = largest[1];

            sign *= -1;
            int i2;
            if (d1 == -1) {
                i2 = i1 - 1;
                p[i1].swap(p[i2]);
                if (i2 == 0 || p[i2 - 1][0] > n1)
                    p[i2][1] = 0;
            } else if (d1 == 1) {
                i2 = i1 + 1;
                p[i1].swap(p[i2]);
                if (i2 == n - 1 || p[i2 + 1][0] > n1)
                    p[i2][1] = 0;
            }

            if (doCopy) {
                aux[0] = p.map!(pi => pi[0]).array;
            } else {
                foreach (immutable i, immutable pi; p)
                    aux[0][i] = pi[0];
            }
            aux[1] = sign;
            result = dg(aux);
            if (result)
                goto END;

            foreach (immutable i3, ref pi; p) {
                immutable n3 = pi[0],
                          d3 = pi[1];
                if (n3 > n1)
                    pi[1] = (i3 < i2) ? 1 : -1;
            }
        }

        END: return result;
    }
}

Spermutations!doCopy spermutations(bool doCopy=true)(in uint n) {
    return typeof(return)(n);
}

version (permutations_by_swapping1) {
    void main() {
        import std.stdio;
        foreach (immutable n; [3, 4]) {
            writefln("\nPermutations and sign of %d items", n);
            foreach (const tp; n.spermutations)
                writefln("Perm: %s  Sign: %2d", tp[]);
        }
    }
}

Compile with version=permutations_by_swapping1 to see the demo output.

Output:
Permutations and sign of 3 items
Perm: [0, 1, 2]  Sign:  1
Perm: [0, 2, 1]  Sign: -1
Perm: [2, 0, 1]  Sign:  1
Perm: [2, 1, 0]  Sign: -1
Perm: [1, 2, 0]  Sign:  1
Perm: [1, 0, 2]  Sign: -1

Permutations and sign of 4 items
Perm: [0, 1, 2, 3]  Sign:  1
Perm: [0, 1, 3, 2]  Sign: -1
Perm: [0, 3, 1, 2]  Sign:  1
Perm: [3, 0, 1, 2]  Sign: -1
Perm: [3, 0, 2, 1]  Sign:  1
Perm: [0, 3, 2, 1]  Sign: -1
Perm: [0, 2, 3, 1]  Sign:  1
Perm: [0, 2, 1, 3]  Sign: -1
Perm: [2, 0, 1, 3]  Sign:  1
Perm: [2, 0, 3, 1]  Sign: -1
Perm: [2, 3, 0, 1]  Sign:  1
Perm: [3, 2, 0, 1]  Sign: -1
Perm: [3, 2, 1, 0]  Sign:  1
Perm: [2, 3, 1, 0]  Sign: -1
Perm: [2, 1, 3, 0]  Sign:  1
Perm: [2, 1, 0, 3]  Sign: -1
Perm: [1, 2, 0, 3]  Sign:  1
Perm: [1, 2, 3, 0]  Sign: -1
Perm: [1, 3, 2, 0]  Sign:  1
Perm: [3, 1, 2, 0]  Sign: -1
Perm: [3, 1, 0, 2]  Sign:  1
Perm: [1, 3, 0, 2]  Sign: -1
Perm: [1, 0, 3, 2]  Sign:  1
Perm: [1, 0, 2, 3]  Sign: -1

Recursive Version

Translation of: Python
import std.algorithm, std.array, std.typecons, std.range;

auto sPermutations(in uint n) pure nothrow @safe {
    static immutable(int[])[] inner(in int items) pure nothrow @safe {
        if (items <= 0)
            return [[]];
        typeof(return) r;
        foreach (immutable i, immutable item; inner(items - 1)) {
            //r.put((i % 2 ? iota(item.length.signed, -1, -1) :
            //               iota(item.length + 1))
            //      .map!(i => item[0 .. i] ~ (items - 1) ~ item[i .. $]));
            immutable f = (in size_t i) pure nothrow @safe =>
                item[0 .. i] ~ (items - 1) ~ item[i .. $];
            r ~= (i % 2) ?
                 //iota(item.length.signed, -1, -1).map!f.array :
                 iota(item.length + 1).retro.map!f.array :
                 iota(item.length + 1).map!f.array;
        }
        return r;
    }

    return inner(n).zip([1, -1].cycle);
}

void main() {
    import std.stdio;
    foreach (immutable n; [2, 3, 4]) {
        writefln("Permutations and sign of %d items:", n);
        foreach (immutable tp; n.sPermutations)
            writefln("  %s Sign: %2d", tp[]);
        writeln;
    }
}
Output:
Permutations and sign of 2 items:
  [1, 0] Sign:  1
  [0, 1] Sign: -1

Permutations and sign of 3 items:
  [2, 1, 0] Sign:  1
  [1, 2, 0] Sign: -1
  [1, 0, 2] Sign:  1
  [0, 1, 2] Sign: -1
  [0, 2, 1] Sign:  1
  [2, 0, 1] Sign: -1

Permutations and sign of 4 items:
  [3, 2, 1, 0] Sign:  1
  [2, 3, 1, 0] Sign: -1
  [2, 1, 3, 0] Sign:  1
  [2, 1, 0, 3] Sign: -1
  [1, 2, 0, 3] Sign:  1
  [1, 2, 3, 0] Sign: -1
  [1, 3, 2, 0] Sign:  1
  [3, 1, 2, 0] Sign: -1
  [3, 1, 0, 2] Sign:  1
  [1, 3, 0, 2] Sign: -1
  [1, 0, 3, 2] Sign:  1
  [1, 0, 2, 3] Sign: -1
  [0, 1, 2, 3] Sign:  1
  [0, 1, 3, 2] Sign: -1
  [0, 3, 1, 2] Sign:  1
  [3, 0, 1, 2] Sign: -1
  [3, 0, 2, 1] Sign:  1
  [0, 3, 2, 1] Sign: -1
  [0, 2, 3, 1] Sign:  1
  [0, 2, 1, 3] Sign: -1
  [2, 0, 1, 3] Sign:  1
  [2, 0, 3, 1] Sign: -1
  [2, 3, 0, 1] Sign:  1
  [3, 2, 0, 1] Sign: -1

Dart

Translation of: Java
void main() {
  List<int> array = List.generate(4, (i) => i);
  HeapsAlgorithm algorithm = HeapsAlgorithm();
  algorithm.recursive(array);
  print('');
  algorithm.loop(array);
}

class HeapsAlgorithm {
  void recursive(List array) {
    _recursive(array, array.length, true);
  }

  void _recursive(List array, int n, bool plus) {
    if (n == 1) {
      _output(array, plus);
    } else {
      for (int i = 0; i < n; i++) {
        _recursive(array, n - 1, i == 0);
        _swap(array, n % 2 == 0 ? i : 0, n - 1);
      }
    }
  }

  void _output(List array, bool plus) {
    print(array.toString() + (plus ? ' +1' : ' -1'));
  }

  void _swap(List array, int a, int b) {
    var temp = array[a];
    array[a] = array[b];
    array[b] = temp;
  }

  void loop(List array) {
    _loop(array, array.length);
  }

  void _loop(List array, int n) {
    List<int> c = List.filled(n, 0);
    _output(array, true);
    bool plus = false;
    int i = 0;
    while (i < n) {
      if (c[i] < i) {
        if (i % 2 == 0) {
          _swap(array, 0, i);
        } else {
          _swap(array, c[i], i);
        }
        _output(array, plus);
        plus = !plus;
        c[i]++;
        i = 0;
      } else {
        c[i] = 0;
        i++;
      }
    }
  }
}
Output:
[0, 1, 2, 3] +1
[1, 0, 2, 3] -1
[2, 0, 1, 3] +1
[0, 2, 1, 3] -1
[1, 2, 0, 3] +1
[2, 1, 0, 3] -1
[3, 1, 2, 0] +1
[1, 3, 2, 0] -1
[2, 3, 1, 0] +1
[3, 2, 1, 0] -1
[1, 2, 3, 0] +1
[2, 1, 3, 0] -1
[3, 0, 2, 1] +1
[0, 3, 2, 1] -1
[2, 3, 0, 1] +1
[3, 2, 0, 1] -1
[0, 2, 3, 1] +1
[2, 0, 3, 1] -1
[3, 0, 1, 2] +1
[0, 3, 1, 2] -1
[1, 3, 0, 2] +1
[3, 1, 0, 2] -1
[0, 1, 3, 2] +1
[1, 0, 3, 2] -1

[3, 0, 1, 2] +1
[0, 3, 1, 2] -1
[1, 3, 0, 2] +1
[3, 1, 0, 2] -1
[0, 1, 3, 2] +1
[1, 0, 3, 2] -1
[2, 0, 3, 1] +1
[0, 2, 3, 1] -1
[3, 2, 0, 1] +1
[2, 3, 0, 1] -1
[0, 3, 2, 1] +1
[3, 0, 2, 1] -1
[3, 1, 2, 0] +1
[1, 3, 2, 0] -1
[2, 3, 1, 0] +1
[3, 2, 1, 0] -1
[1, 2, 3, 0] +1
[2, 1, 3, 0] -1
[2, 1, 0, 3] +1
[1, 2, 0, 3] -1
[0, 2, 1, 3] +1
[2, 0, 1, 3] -1
[1, 0, 2, 3] +1
[0, 1, 2, 3] -1

Delphi

Works with: Delphi version 6.0


{These routines would normally be in a separate library; they are presented here for clarity}


{Permutator based on the Johnson and Trotter algorithm.}
{Which only permutates by swapping a pair of elements at a time}
{object steps through all permutation of array items}
{Zero-Based = True = 0..Permutions-1 False = 1..Permutaions}
{Permutation set on "Create(Size)" or by "Permutations" property}
{Permutation are contained in the array "Indices"}

type TDirection = (drLeftToRight,drRightToLeft);
type TDirArray = array of TDirection;


type TJTPermutator = class(TObject)
 private
  Dir: TDirArray;
  FZeroBased: boolean;
  FBase: integer;
  FPermutations: integer;
  procedure SetZeroBased(const Value: boolean);
  procedure SetPermutations(const Value: integer);
 protected
   FMax: integer;
 public
  NextCount: Integer;
  Indices: TIntegerDynArray;
  constructor Create(Size: integer);
  procedure Reset;
  function Next: boolean;
  property ZeroBased: boolean read FZeroBased write SetZeroBased;
  property Permutations: integer read FPermutations write SetPermutations;
 end;


{==============================================================================}

function Fact(N: integer): integer;
{Get factorial of N}
var I: integer;
begin
Result:=1;
for I:=1 to N do Result:=Result * I;
end;


procedure SwapIntegers(var A1,A2: integer);
{Swap integer arguments}
var T: integer;
begin
T:=A1; A1:=A2; A2:=T;
end;


procedure TJTPermutator.Reset;
var I: integer;
begin
{ Preset items 0..n-1 or 1..n depending on base}
for I:=0 to High(Indices) do Indices[I]:=I + FBase;
{ initially all directions are set to RIGHT TO LEFT }
for I:=0 to High(Indices) do Dir[I]:=drRightToLeft;
NextCount:=0;
end;


procedure TJTPermutator.SetPermutations(const Value: integer);
begin
if FPermutations<>Value then
	begin
	FPermutations := Value;
	SetLength(Indices,Value);
	SetLength(Dir,Value);
	Reset;
	end;
end;



constructor TJTPermutator.Create(Size: integer);
begin
ZeroBased:=True;
Permutations:=Size;
Reset;
end;



procedure TJTPermutator.SetZeroBased(const Value: boolean);
begin
if FZeroBased<>Value then
	begin
	FZeroBased := Value;
	if Value then FBase:=0
	else FBase:=1;
	Reset;
	end;
end;


function TJTPermutator.Next: boolean;
{Step to next permutation}
{Returns true when sequence completed}
var Mobile,Pos,I: integer;
var S: string;

	function FindLargestMoble(Mobile: integer): integer;
	{Find position of largest mobile integer in A}
	var I: integer;
	begin
	for I:=0 to  High(Indices) do
	 if Indices[I] = Mobile then
		begin
		Result:=I + 1;
		exit;
		end;
	Result:=-1;
	end;


	function GetMobile: integer;
	{ find the largest mobile integer.}
	var LastMobile, Mobile: integer;
	var I: integer;
	begin
	LastMobile:= 0; Mobile:= 0;
	for I:=0 to High(Indices) do
		begin
		{ direction 0 represents RIGHT TO LEFT.}
		if (Dir[Indices[I] - 1] = drRightToLeft) and (I<>0) then
			begin
			if (Indices[I] > Indices[I - 1]) and (Indices[I] > LastMobile) then
				begin
				Mobile:=Indices[I];
				LastMobile:=Mobile;
				end;
			end;

	        { direction 1 represents LEFT TO RIGHT.}
	        if (dir[Indices[I] - 1] = drLeftToRight) and (i<>(Length(Indices) - 1)) then
			begin
			if (Indices[I] > Indices[I + 1]) and (Indices[I] > LastMobile) then
				begin
				Mobile:=Indices[I];
				LastMobile:=Mobile;
				end;
			end;
		end;

	if (Mobile = 0) and (LastMobile = 0) then Result:=0
	else Result:=Mobile;
	end;



begin
Inc(NextCount);
Result:=NextCount>=Fact(Length(Indices));
if Result then
	begin
	Reset;
	exit;
	end;
Mobile:=GetMobile;
Pos:=FindLargestMoble(Mobile);

{ Swap elements according to the direction in Dir}
if (Dir[Indices[pos - 1] - 1] = drRightToLeft) then SwapIntegers(Indices[Pos - 1], Indices[Pos - 2])
else if (dir[Indices[pos - 1] - 1] = drLeftToRight) then SwapIntegers(Indices[Pos], Indices[Pos - 1]);

{ changing the directions for elements}
{ greater than largest Mobile integer.}
for I:=0 to High(Indices) do
 if Indices[I] > Mobile then
	begin
	if Dir[Indices[I] - 1] = drLeftToRight then Dir[Indices[I] - 1]:=drRightToLeft
	else if (Dir[Indices[i] - 1] = drRightToLeft) then Dir[Indices[I] - 1]:=drLeftToRight;
        end;
end;


{==============================================================================}




function GetPermutationStr(PM: TJTPermutator): string;
var I: integer;
begin
Result:=Format('%2d - [',[PM.NextCount+1]);
for I:=0 to High(PM.Indices) do Result:=Result+IntToStr(PM.Indices[I]);
Result:=Result+'] Sign: ';
if (PM.NextCount and 1)=0 then Result:=Result+'+1'
else Result:=Result+'-1';
end;



procedure SwapPermutations(Memo: TMemo);
var PM: TJTPermutator;
begin
PM:=TJTPermutator.Create(3);
try
repeat Memo.Lines.Add(GetPermutationStr(PM))
until PM.Next;
Memo.Lines.Add('');

PM.Permutations:=4;
repeat Memo.Lines.Add(GetPermutationStr(PM))
until PM.Next;
finally PM.Free; end;
end;
Output:
 1 - [012] Sign: +1
 2 - [021] Sign: -1
 3 - [201] Sign: +1
 4 - [210] Sign: -1
 5 - [120] Sign: +1
 6 - [102] Sign: -1

 1 - [0123] Sign: +1
 2 - [0132] Sign: -1
 3 - [0312] Sign: +1
 4 - [3012] Sign: -1
 5 - [3021] Sign: +1
 6 - [0321] Sign: -1
 7 - [0231] Sign: +1
 8 - [0213] Sign: -1
 9 - [2013] Sign: +1
10 - [2031] Sign: -1
11 - [2301] Sign: +1
12 - [3201] Sign: -1
13 - [3210] Sign: +1
14 - [2310] Sign: -1
15 - [2130] Sign: +1
16 - [2103] Sign: -1
17 - [1203] Sign: +1
18 - [1230] Sign: -1
19 - [1320] Sign: +1
20 - [3120] Sign: -1
21 - [3102] Sign: +1
22 - [1302] Sign: -1
23 - [1032] Sign: +1
24 - [1023] Sign: -1

Elapsed Time: 60.734 ms.


EasyLang

# Heap's Algorithm
sig = 1
proc generate k . ar[] .
   if k = 1
      print ar[] & "  " & sig
      sig = -sig
      return
   .
   generate k - 1 ar[]
   for i to k - 1
      if k mod 2 = 0
         swap ar[i] ar[k]
      else
         swap ar[1] ar[k]
      .
      generate k - 1 ar[]
   .
.
ar[] = [ 1 2 3 ]
generate len ar[] ar[]
Output:
[ 1 2 3 ]  1
[ 2 1 3 ]  -1
[ 3 1 2 ]  1
[ 1 3 2 ]  -1
[ 2 3 1 ]  1
[ 3 2 1 ]  -1

EchoLisp

The function (in-permutations n) returns a stream which delivers permutations according to the Steinhaus–Johnson–Trotter algorithm.

(lib 'list)

(for/fold (sign 1) ((σ (in-permutations 4)) (count 100)) 
    (printf "perm: %a count:%4d sign:%4d" σ count sign) (* sign -1))

perm: (0 1 2 3) count:   0 sign:   1
perm: (0 1 3 2) count:   1 sign:  -1
perm: (0 3 1 2) count:   2 sign:   1
perm: (3 0 1 2) count:   3 sign:  -1
perm: (3 0 2 1) count:   4 sign:   1
perm: (0 3 2 1) count:   5 sign:  -1
perm: (0 2 3 1) count:   6 sign:   1
perm: (0 2 1 3) count:   7 sign:  -1
perm: (2 0 1 3) count:   8 sign:   1
perm: (2 0 3 1) count:   9 sign:  -1
perm: (2 3 0 1) count:  10 sign:   1
perm: (3 2 0 1) count:  11 sign:  -1
perm: (3 2 1 0) count:  12 sign:   1
perm: (2 3 1 0) count:  13 sign:  -1
perm: (2 1 3 0) count:  14 sign:   1
perm: (2 1 0 3) count:  15 sign:  -1
perm: (1 2 0 3) count:  16 sign:   1
perm: (1 2 3 0) count:  17 sign:  -1
perm: (1 3 2 0) count:  18 sign:   1
perm: (3 1 2 0) count:  19 sign:  -1
perm: (3 1 0 2) count:  20 sign:   1
perm: (1 3 0 2) count:  21 sign:  -1
perm: (1 0 3 2) count:  22 sign:   1
perm: (1 0 2 3) count:  23 sign:  -1

Elixir

Translation of: Ruby
defmodule Permutation do
  def by_swap(n) do
    p = Enum.to_list(0..-n) |> List.to_tuple
    by_swap(n, p, 1)
  end
  
  defp by_swap(n, p, s) do
    IO.puts "Perm: #{inspect for i <- 1..n, do: abs(elem(p,i))}  Sign: #{s}"
    k = 0 |> step_up(n, p) |> step_down(n, p)
    if k > 0 do
      pk = elem(p,k)
      i = if pk>0, do: k+1, else: k-1
      p = Enum.reduce(1..n, p, fn i,acc ->
        if abs(elem(p,i)) > abs(pk), do: put_elem(acc, i, -elem(acc,i)), else: acc
      end)
      pi = elem(p,i)
      p = put_elem(p,i,pk) |> put_elem(k,pi)            # swap
      by_swap(n, p, -s)
    end
  end
  
  defp step_up(k, n, p) do
    Enum.reduce(2..n, k, fn i,acc ->
      if elem(p,i)<0 and abs(elem(p,i))>abs(elem(p,i-1)) and abs(elem(p,i))>abs(elem(p,acc)),
        do: i, else: acc 
    end)
  end
  
  defp step_down(k, n, p) do
    Enum.reduce(1..n-1, k, fn i,acc ->
      if elem(p,i)>0 and abs(elem(p,i))>abs(elem(p,i+1)) and abs(elem(p,i))>abs(elem(p,acc)),
        do: i, else: acc 
    end)
  end
end

Enum.each(3..4, fn n ->
  Permutation.by_swap(n)
  IO.puts ""
end)
Output:
Perm: [1, 2, 3]  Sign: 1
Perm: [1, 3, 2]  Sign: -1
Perm: [3, 1, 2]  Sign: 1
Perm: [3, 2, 1]  Sign: -1
Perm: [2, 3, 1]  Sign: 1
Perm: [2, 1, 3]  Sign: -1

Perm: [1, 2, 3, 4]  Sign: 1
Perm: [1, 2, 4, 3]  Sign: -1
Perm: [1, 4, 2, 3]  Sign: 1
Perm: [4, 1, 2, 3]  Sign: -1
Perm: [4, 1, 3, 2]  Sign: 1
Perm: [1, 4, 3, 2]  Sign: -1
Perm: [1, 3, 4, 2]  Sign: 1
Perm: [1, 3, 2, 4]  Sign: -1
Perm: [3, 1, 2, 4]  Sign: 1
Perm: [3, 1, 4, 2]  Sign: -1
Perm: [3, 4, 1, 2]  Sign: 1
Perm: [4, 3, 1, 2]  Sign: -1
Perm: [4, 3, 2, 1]  Sign: 1
Perm: [3, 4, 2, 1]  Sign: -1
Perm: [3, 2, 4, 1]  Sign: 1
Perm: [3, 2, 1, 4]  Sign: -1
Perm: [2, 3, 1, 4]  Sign: 1
Perm: [2, 3, 4, 1]  Sign: -1
Perm: [2, 4, 3, 1]  Sign: 1
Perm: [4, 2, 3, 1]  Sign: -1
Perm: [4, 2, 1, 3]  Sign: 1
Perm: [2, 4, 1, 3]  Sign: -1
Perm: [2, 1, 4, 3]  Sign: 1
Perm: [2, 1, 3, 4]  Sign: -1

F#

See [2] for an example using this module

(*Implement Johnson-Trotter algorithm
  Nigel Galloway January 24th 2017*)
module Ring
let PlainChanges (N:'n[]) = seq{
  let gn  = [|for n in N -> 1|]
  let ni  = [|for n in N -> 0|]
  let gel = Array.length(N)-1
  yield N
  let rec _Ni g e l = seq{
    match (l,g) with
    |_ when l<0   -> gn.[g] <- -gn.[g]; yield! _Ni (g-1) e (ni.[g-1] + gn.[g-1])
    |(1,0)        -> ()
    |_ when l=g+1 -> gn.[g] <- -gn.[g]; yield! _Ni (g-1) (e+1) (ni.[g-1] + gn.[g-1])
    |_ -> let n = N.[g-ni.[g]+e];
          N.[g-ni.[g]+e] <- N.[g-l+e]; N.[g-l+e] <- n; yield N
          ni.[g] <- l; yield! _Ni gel 0 (ni.[gel] + gn.[gel])}
  yield! _Ni gel 0 1
}

A little code for the purpose of this task demonstrating the algorithm

for n in Ring.PlainChanges [|1;2;3;4|] do printfn "%A" n
Output:
[|1; 2; 3; 4|]
[|1; 2; 4; 3|]
[|1; 4; 2; 3|]
[|4; 1; 2; 3|]
[|4; 1; 3; 2|]
[|1; 4; 3; 2|]
[|1; 3; 4; 2|]
[|1; 3; 2; 4|]
[|3; 1; 2; 4|]
[|3; 1; 4; 2|]
[|3; 4; 1; 2|]
[|4; 3; 1; 2|]
[|4; 3; 2; 1|]
[|3; 4; 2; 1|]
[|3; 2; 4; 1|]
[|3; 2; 1; 4|]
[|2; 3; 1; 4|]
[|2; 3; 4; 1|]
[|2; 4; 3; 1|]
[|4; 2; 3; 1|]
[|4; 2; 1; 3|]
[|2; 4; 1; 3|]
[|2; 1; 4; 3|]
[|2; 1; 3; 4|]
v

Forth

Works with: gforth version 0.7.9_20170308
Translation of: BBC BASIC
S" fsl-util.fs" REQUIRED
S" fsl/dynmem.seq" REQUIRED

cell darray p{

: sgn
  DUP 0 > IF
    DROP 1
  ELSE 0 < IF
    -1
  ELSE
    0
  THEN THEN ;
: arr-swap {: addr1 addr2 | tmp -- :}
  addr1 @ TO tmp
  addr2 @ addr1 !
  tmp addr2 ! ;
: perms {: n xt | my-i k s -- :}
  & p{ n 1+ }malloc malloc-fail? ABORT" perms :: out of memory"
  0 p{ 0 } !
  n 1+ 1 DO
    I NEGATE p{ I } !
  LOOP
  1 TO s
  BEGIN
    1 n 1+ DO
      p{ I } @ ABS
    -1 +LOOP
    n 1+ s xt EXECUTE
    0 TO k
    n 1+ 2 DO
      p{ I } @ 0 < ( flag )
      p{ I } @ ABS  p{ I 1- } @ ABS  > ( flag flag )
      p{ I } @ ABS p{ k } @ ABS > ( flag flag flag )
      AND AND IF
        I TO k
      THEN
    LOOP
    n 1 DO
      p{ I } @ 0 > ( flag )
      p{ I } @ ABS  p{ I 1+ } @ ABS  > ( flag flag )
      p{ I } @ ABS  p{ k } @ ABS  > ( flag flag flag )
      AND AND IF
        I TO k
      THEN
    LOOP
    k IF
      n 1+ 1 DO
        p{ I } @ ABS  p{ k } @ ABS  > IF
          p{ I } @ NEGATE p{ I } !
        THEN
      LOOP
      p{ k } @ sgn k + TO my-i
      p{ k } p{ my-i } arr-swap
      s NEGATE TO s
    THEN
  k 0 = UNTIL ;
: .perm ( p0 p1 p2 ... pn n s )
  >R
  ." Perm: [ "
  1 DO
    . SPACE
  LOOP
  R> ." ] Sign: " . CR ;

3 ' .perm perms CR
4 ' .perm perms

FreeBASIC

Translation of: BBC BASIC
' version 31-03-2017
' compile with: fbc -s console

Sub perms(n As ULong)

    Dim As Long p(n), i, k, s = 1

    For i = 1 To n
        p(i) = -i
    Next

    Do
        Print "Perm: [ ";
        For i = 1 To n
            Print Abs(p(i)); " ";
        Next
        Print "] Sign: "; s

        k = 0
        For i = 2 To n
            If p(i) < 0 Then
                If Abs(p(i)) > Abs(p(i -1)) Then
                    If Abs(p(i)) > Abs(p(k)) Then k = i
                End If
            End If
        Next

        For i = 1 To n -1
            If p(i) > 0 Then
                If Abs(p(i)) > Abs(p(i +1)) Then
                    If Abs(p(i)) > Abs(p(k)) Then k = i
                End If
            End If
        Next

        If k Then
            For  i = 1 To n
                If Abs(p(i)) > Abs(p(k)) Then p(i) = -p(i)
            Next
            i = k + Sgn(p(k))
            Swap p(k), p(i)
            s = -s
        End If

    Loop Until k = 0

End Sub

' ------=< MAIN >=------

perms(3)
print
perms(4)

' empty keyboard buffer
While Inkey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End
Output:
output is edited to show results side by side
       Perm: [  1  2  3 ] Sign:  1         Perm: [  1  2  3  4 ] Sign:  1
       Perm: [  1  3  2 ] Sign: -1         Perm: [  1  2  4  3 ] Sign: -1
       Perm: [  3  1  2 ] Sign:  1         Perm: [  1  4  2  3 ] Sign:  1
       Perm: [  3  2  1 ] Sign: -1         Perm: [  4  1  2  3 ] Sign: -1
       Perm: [  2  3  1 ] Sign:  1         Perm: [  4  1  3  2 ] Sign:  1
       Perm: [  2  1  3 ] Sign: -1         Perm: [  1  4  3  2 ] Sign: -1
                                           Perm: [  1  2  3  4 ] Sign:  1
                                           Perm: [  1  2  4  3 ] Sign: -1
                                           Perm: [  1  4  2  3 ] Sign:  1
                                           Perm: [  4  1  2  3 ] Sign: -1
                                           Perm: [  4  1  3  2 ] Sign:  1
                                           Perm: [  1  4  3  2 ] Sign: -1
                                           Perm: [  1  3  4  2 ] Sign:  1
                                           Perm: [  1  3  2  4 ] Sign: -1
                                           Perm: [  3  1  2  4 ] Sign:  1
                                           Perm: [  3  1  4  2 ] Sign: -1
                                           Perm: [  3  4  1  2 ] Sign:  1
                                           Perm: [  4  3  1  2 ] Sign: -1
                                           Perm: [  4  3  2  1 ] Sign:  1
                                           Perm: [  3  4  2  1 ] Sign: -1
                                           Perm: [  3  2  4  1 ] Sign:  1
                                           Perm: [  3  2  1  4 ] Sign: -1
                                           Perm: [  2  3  1  4 ] Sign:  1
                                           Perm: [  2  3  4  1 ] Sign: -1
                                           Perm: [  2  4  3  1 ] Sign:  1
                                           Perm: [  4  2  3  1 ] Sign: -1
                                           Perm: [  4  2  1  3 ] Sign:  1
                                           Perm: [  2  4  1  3 ] Sign: -1
                                           Perm: [  2  1  4  3 ] Sign:  1
                                           Perm: [  2  1  3  4 ] Sign: -1

Go

package permute

// Iter takes a slice p and returns an iterator function.  The iterator
// permutes p in place and returns the sign.  After all permutations have
// been generated, the iterator returns 0 and p is left in its initial order.
func Iter(p []int) func() int {
    f := pf(len(p))
    return func() int {
        return f(p)
    }
}

// Recursive function used by perm, returns a chain of closures that
// implement a loopless recursive SJT.
func pf(n int) func([]int) int {
    sign := 1
    switch n {
    case 0, 1:
        return func([]int) (s int) {
            s = sign
            sign = 0
            return
        }
    default:
        p0 := pf(n - 1)
        i := n
        var d int
        return func(p []int) int {
            switch {
            case sign == 0:
            case i == n:
                i--
                sign = p0(p[:i])
                d = -1
            case i == 0:
                i++
                sign *= p0(p[1:])
                d = 1
                if sign == 0 {
                    p[0], p[1] = p[1], p[0]
                }
            default:
                p[i], p[i-1] = p[i-1], p[i]
                sign = -sign
                i += d
            }
            return sign
        }
    }
}
package main

import (
    "fmt"
    "permute"
)

func main() {
    p := []int{11, 22, 33}
    i := permute.Iter(p)
    for sign := i(); sign != 0; sign = i() {
        fmt.Println(p, sign)
    }
}
Output:
[11 22 33] 1
[11 33 22] -1
[33 11 22] 1
[33 22 11] -1
[22 33 11] 1
[22 11 33] -1

Haskell

sPermutations :: [a] -> [([a], Int)]
sPermutations = flip zip (cycle [-1, 1]) . foldr aux [[]]
  where
    aux x items = do
      (f, item) <- zip (repeat id) items
      f (insertEv x item)
    insertEv x [] = [[x]]
    insertEv x l@(y:ys) = (x : l) : ((y :) <$> insertEv x ys)

main :: IO ()
main = do
  putStrLn "3 items:"
  mapM_ print $ sPermutations [1 .. 3]
  putStrLn "\n4 items:"
  mapM_ print $ sPermutations [1 .. 4]
Output:
3 items:
([1,2,3],-1)
([2,1,3],1)
([2,3,1],-1)
([1,3,2],1)
([3,1,2],-1)
([3,2,1],1)

4 items:
([1,2,3,4],-1)
([2,1,3,4],1)
([2,3,1,4],-1)
([2,3,4,1],1)
([1,3,2,4],-1)
([3,1,2,4],1)
([3,2,1,4],-1)
([3,2,4,1],1)
([1,3,4,2],-1)
([3,1,4,2],1)
([3,4,1,2],-1)
([3,4,2,1],1)
([1,2,4,3],-1)
([2,1,4,3],1)
([2,4,1,3],-1)
([2,4,3,1],1)
([1,4,2,3],-1)
([4,1,2,3],1)
([4,2,1,3],-1)
([4,2,3,1],1)
([1,4,3,2],-1)
([4,1,3,2],1)
([4,3,1,2],-1)
([4,3,2,1],1)

Icon and Unicon

Works in both languages.

Translation of: Python
procedure main(A)
    every write("Permutations of length ",n := !A) do
       every p := permute(n) do write("\t",showList(p[1])," -> ",right(p[2],2))
end
 
procedure permute(n)
    items := [[]]
    every (j := 1 to n, new_items := []) do {
        every item := items[i := 1 to *items] do {
            if *item = 0 then put(new_items, [j])
            else if i%2 = 0 then
                every k := 1 to *item+1 do {
                    new_item := item[1:k] ||| [j] ||| item[k:0]
                    put(new_items, new_item)
                    }
            else
                every k := *item+1 to 1 by -1 do {
                    new_item := item[1:k] ||| [j] ||| item[k:0]
                    put(new_items, new_item)
                    }
            }
       items := new_items
       }
    suspend (i := 0, [!items, if (i+:=1)%2 = 0 then 1 else -1])
end

procedure showList(A)
    every (s := "[") ||:= image(!A)||", "
    return s[1:-2]||"]"
end

Sample run:

->pbs 3 4
Permutations of length 3
        [1, 2, 3] -> -1
        [1, 3, 2] ->  1
        [3, 1, 2] -> -1
        [3, 2, 1] ->  1
        [2, 3, 1] -> -1
        [2, 1, 3] ->  1
Permutations of length 4
        [1, 2, 3, 4] -> -1
        [1, 2, 4, 3] ->  1
        [1, 4, 2, 3] -> -1
        [4, 1, 2, 3] ->  1
        [4, 1, 3, 2] -> -1
        [1, 4, 3, 2] ->  1
        [1, 3, 4, 2] -> -1
        [1, 3, 2, 4] ->  1
        [3, 1, 2, 4] -> -1
        [3, 1, 4, 2] ->  1
        [3, 4, 1, 2] -> -1
        [4, 3, 1, 2] ->  1
        [4, 3, 2, 1] -> -1
        [3, 4, 2, 1] ->  1
        [3, 2, 4, 1] -> -1
        [3, 2, 1, 4] ->  1
        [2, 3, 1, 4] -> -1
        [2, 3, 4, 1] ->  1
        [2, 4, 3, 1] -> -1
        [4, 2, 3, 1] ->  1
        [4, 2, 1, 3] -> -1
        [2, 4, 1, 3] ->  1
        [2, 1, 4, 3] -> -1
        [2, 1, 3, 4] ->  1
->

J

J has a built in mechanism for representing permutations for selecting a permutation of a given length with an integer, but this mechanism does not seem to have an obvious mapping to Steinhaus–Johnson–Trotter. Perhaps someone with a sufficiently deep view of the subject of permutations can find a direct mapping?

Meanwhile, here's an inductive approach, using negative integers to look left and positive integers to look right:

bfsjt0=: _1 - i.
lookingat=: 0 >. <:@# <. i.@# + * 
next=: | >./@:* | > | {~ lookingat
bfsjtn=: (((] <@, ] + *@{~) | i. next) C. ] * _1 ^ next < |)^:(*@next)

Here, bfsjt0 N gives the initial permutation of order N, and bfsjtn^:M bfsjt0 N gives the Mth Steinhaus–Johnson–Trotter permutation of order N. (bf stands for "brute force".)

To convert from the Steinhaus–Johnson–Trotter representation of a permutation to J's representation, use <:@|, or to find J's anagram index of a Steinhaus–Johnson–Trotter representation of a permutation, use A.@:<:@:|

Example use:

   bfsjtn^:(i.!3) bfjt0 3
_1 _2 _3
_1 _3 _2
_3 _1 _2
 3 _2 _1
_2  3 _1
_2 _1  3
   <:@| bfsjtn^:(i.!3) bfjt0 3
0 1 2
0 2 1
2 0 1
2 1 0
1 2 0
1 0 2
   A. <:@| bfsjtn^:(i.!3) bfjt0 3
0 1 4 5 3 2

Here's an example of the Steinhaus–Johnson–Trotter representation of 3 element permutation, with sign (sign is the first column):

   (_1^2|i.!3),. bfsjtn^:(i.!3) bfjt0 3
 1 _1 _2 _3
_1 _1 _3 _2
 1 _3 _1 _2
_1  3 _2 _1
 1 _2  3 _1
_1 _2 _1  3

Alternatively, J defines C.!.2 as the parity of a permutation:

   (,.~C.!.2)<:| bfsjtn^:(i.!3) bfjt0 3
 1 0 1 2
_1 0 2 1
 1 2 0 1
_1 2 1 0
 1 1 2 0
_1 1 0 2

Recursive Implementation

This is based on the python recursive implementation:

rsjt=: 3 :0
  if. 2>y do. i.2#y
  else.  ((!y)$(,~|.)-.=i.y)#inv!.(y-1)"1 y#rsjt y-1
  end.
)

Example use (here, prefixing each row with its parity):

   (,.~ C.!.2) rsjt 3
 1 0 1 2
_1 0 2 1
 1 2 0 1
_1 2 1 0
 1 1 2 0
_1 1 0 2

Java

Heap's Algorithm, recursive and looping implementations

package org.rosettacode.java;

import java.util.Arrays;
import java.util.stream.IntStream;

public class HeapsAlgorithm {

	public static void main(String[] args) {
		Object[] array = IntStream.range(0, 4)
				.boxed()
				.toArray();
		HeapsAlgorithm algorithm = new HeapsAlgorithm();
		algorithm.recursive(array);
		System.out.println();
		algorithm.loop(array);
	}

	void recursive(Object[] array) {
		recursive(array, array.length, true);
	}

	void recursive(Object[] array, int n, boolean plus) {
		if (n == 1) {
			output(array, plus);
		} else {
			for (int i = 0; i < n; i++) {
				recursive(array, n - 1, i == 0);
				swap(array, n % 2 == 0 ? i : 0, n - 1);
			}
		}
	}

	void output(Object[] array, boolean plus) {
		System.out.println(Arrays.toString(array) + (plus ? " +1" : " -1"));
	}

	void swap(Object[] array, int a, int b) {
		Object o = array[a];
		array[a] = array[b];
		array[b] = o;
	}

	void loop(Object[] array) {
		loop(array, array.length);
	}

	void loop(Object[] array, int n) {
		int[] c = new int[n];
		output(array, true);
		boolean plus = false;
		for (int i = 0; i < n; ) {
			if (c[i] < i) {
				if (i % 2 == 0) {
					swap(array, 0, i);
				} else {
					swap(array, c[i], i);
				}
				output(array, plus);
				plus = !plus;
				c[i]++;
				i = 0;
			} else {
				c[i] = 0;
				i++;
			}
		}
	}
}
Output:
[0, 1, 2, 3] +1
[1, 0, 2, 3] -1
[2, 0, 1, 3] +1
[0, 2, 1, 3] -1
[1, 2, 0, 3] +1
[2, 1, 0, 3] -1
[3, 1, 2, 0] +1
[1, 3, 2, 0] -1
[2, 3, 1, 0] +1
[3, 2, 1, 0] -1
[1, 2, 3, 0] +1
[2, 1, 3, 0] -1
[3, 0, 2, 1] +1
[0, 3, 2, 1] -1
[2, 3, 0, 1] +1
[3, 2, 0, 1] -1
[0, 2, 3, 1] +1
[2, 0, 3, 1] -1
[3, 0, 1, 2] +1
[0, 3, 1, 2] -1
[1, 3, 0, 2] +1
[3, 1, 0, 2] -1
[0, 1, 3, 2] +1
[1, 0, 3, 2] -1

[3, 0, 1, 2] +1
[0, 3, 1, 2] -1
[1, 3, 0, 2] +1
[3, 1, 0, 2] -1
[0, 1, 3, 2] +1
[1, 0, 3, 2] -1
[2, 0, 3, 1] +1
[0, 2, 3, 1] -1
[3, 2, 0, 1] +1
[2, 3, 0, 1] -1
[0, 3, 2, 1] +1
[3, 0, 2, 1] -1
[3, 1, 2, 0] +1
[1, 3, 2, 0] -1
[2, 3, 1, 0] +1
[3, 2, 1, 0] -1
[1, 2, 3, 0] +1
[2, 1, 3, 0] -1
[2, 1, 0, 3] +1
[1, 2, 0, 3] -1
[0, 2, 1, 3] +1
[2, 0, 1, 3] -1
[1, 0, 2, 3] +1
[0, 1, 2, 3] -1

jq

Works with: jq version 1.4

Based on the ruby version - the sequence is generated by swapping adjacent elements.

"permutations" generates a stream of arrays of the form [par, perm], where "par" is the parity of the permutation "perm" of the input array. This array may contain any JSON entities, which are regarded as distinct.

# The helper function, _recurse, is tail-recursive and therefore in
# versions of jq with TCO (tail call optimization) there is no
# overhead associated with the recursion.

def permutations:
  def abs: if . < 0 then -. else . end;
  def sign: if . < 0 then -1 elif . == 0 then 0 else 1 end;
  def swap(i;j): .[i] as $i | .[i] = .[j] | .[j] = $i;

  # input: [ parity, extendedPermutation]
  def _recurse:
    .[0] as $s | .[1] as $p | (($p | length) -1) as $n
    | [ $s, ($p[1:] | map(abs)) ],
      (reduce range(2; $n+1) as $i
         (0;
          if $p[$i] < 0 and -($p[$i]) > ($p[$i-1]|abs) and -($p[$i]) > ($p[.]|abs)
          then $i 
          else .
          end)) as $k
      | (reduce range(1; $n) as $i
           ($k;
            if $p[$i] > 0 and $p[$i] > ($p[$i+1]|abs) and $p[$i] > ($p[.]|abs)
            then $i 
            else .
            end)) as $k
      | if $k == 0 then empty
        else (reduce range(1; $n) as $i
	       ($p;
                if (.[$i]|abs) > (.[$k]|abs) then .[$i] *= -1 
                else .
                end )) as $p
        | ($k + ($p[$k]|sign)) as $i
        | ($p | swap($i; $k)) as $p
        | [ -($s), $p ] | _recurse
        end ;

  . as $in
  | length as $n
  | (reduce range(0; $n+1) as $i ([]; . + [ -$i ])) as $p
  # recurse state: [$s, $p]
  | [ 1, $p] | _recurse
  | .[1] as $p
  | .[1] = reduce range(0; $n) as $i ([]; . + [$in[$p[$i]  - 1]]) ;

def count(stream): reduce stream as $x (0; .+1);

Examples:

(["a", "b", "c"] | permutations),
"There are \(count( [range(1;6)] | permutations )) permutations of 5 items."
Output:
$ jq -c -n -f Permutations_by_swapping.jq
[1,["a","b","c"]]
[-1,["a","c","b"]]
[1,["c","a","b"]]
[-1,["c","b","a"]]
[1,["b","c","a"]]
[-1,["b","a","c"]]

"There are 32 permutations of 5 items."

Julia

Nonrecursive (interative):

function johnsontrottermove!(ints, isleft)
    len = length(ints)
    function ismobile(pos)
        if isleft[pos] && (pos > 1) && (ints[pos-1] < ints[pos])
            return true
        elseif !isleft[pos] && (pos < len) && (ints[pos+1] < ints[pos])
            return true
        end
        false
    end
    function maxmobile()
        arr = [ints[pos] for pos in 1:len if ismobile(pos)]
        if isempty(arr)
            0, 0
        else
            maxmob = maximum(arr)
            maxmob, findfirst(x -> x == maxmob, ints)
        end
    end
    function directedswap(pos)
        tmp = ints[pos]
        tmpisleft = isleft[pos]
        if isleft[pos]
            ints[pos] = ints[pos-1]; ints[pos-1] = tmp
            isleft[pos] = isleft[pos-1]; isleft[pos-1] = tmpisleft
        else
            ints[pos] = ints[pos+1]; ints[pos+1] = tmp
            isleft[pos] = isleft[pos+1]; isleft[pos+1] = tmpisleft
        end
    end
    (moveint, movepos) = maxmobile()
    if movepos > 0
        directedswap(movepos)
        for (i, val) in enumerate(ints)
            if val > moveint
                isleft[i] = !isleft[i]
            end
        end
        ints, isleft, true
    else
        ints, isleft, false
    end
end
function johnsontrotter(low, high)
    ints = collect(low:high)
    isleft = [true for i in ints]
    firstconfig = copy(ints)
    iters = 0
    while true
        iters += 1
        println("$ints $(iters & 1 == 1 ? "+1" : "-1")")
        if johnsontrottermove!(ints, isleft)[3] == false
            break
        end
    end
    println("There were $iters iterations.")
end
johnsontrotter(1,4)

Recursive (note this uses memory of roughtly (n+1)! bytes, where n is the number of elements, in order to store the accumulated permutations in a list, and so the above, iterative solution is to be preferred for numbers of elements over 9 or so):

function johnsontrotter(low, high)
    function permutelevel(vec)
        if length(vec) < 2
            return [vec]
        end
        sequences = []
        endint = vec[end]
        smallersequences = permutelevel(vec[1:end-1])
        leftward = true
        for seq in smallersequences
            for pos in (leftward ? (length(seq)+1:-1:1): (1:length(seq)+1))
                push!(sequences, insert!(copy(seq), pos, endint))
            end
            leftward = !leftward
        end
        sequences
    end
    permutelevel(collect(low:high))
end

for (i, sequence) in enumerate(johnsontrotter(1,4))
    println("""$sequence, $(i & 1 == 1 ? "+1" : "-1")""")
end

Kotlin

This is based on the recursive Java code found at http://introcs.cs.princeton.edu/java/23recursion/JohnsonTrotter.java.html

// version 1.1.2

fun johnsonTrotter(n: Int): Pair<List<IntArray>, List<Int>> {
    val p = IntArray(n) { it }  // permutation
    val q = IntArray(n) { it }  // inverse permutation
    val d = IntArray(n) { -1 }  // direction = 1 or -1
    var sign = 1
    val perms = mutableListOf<IntArray>()
    val signs = mutableListOf<Int>()

    fun permute(k: Int) {
        if (k >= n) {
            perms.add(p.copyOf())
            signs.add(sign)
            sign *= -1
            return
        } 
        permute(k + 1)
        for (i in 0 until k) {
            val z = p[q[k] + d[k]]
            p[q[k]] = z
            p[q[k] + d[k]] = k
            q[z] = q[k]
            q[k] += d[k]
            permute(k + 1)
        }
        d[k] *= -1
    } 

    permute(0)
    return perms to signs
}

fun printPermsAndSigns(perms: List<IntArray>, signs: List<Int>) {
    for ((i, perm) in perms.withIndex()) {
        println("${perm.contentToString()} -> sign = ${signs[i]}")
    }
}

fun main(args: Array<String>) {
    val (perms, signs) = johnsonTrotter(3)
    printPermsAndSigns(perms, signs)
    println()
    val (perms2, signs2) = johnsonTrotter(4)
    printPermsAndSigns(perms2, signs2)
}
Output:
[0, 1, 2] -> sign = 1
[0, 2, 1] -> sign = -1
[2, 0, 1] -> sign = 1
[2, 1, 0] -> sign = -1
[1, 2, 0] -> sign = 1
[1, 0, 2] -> sign = -1

[0, 1, 2, 3] -> sign = 1
[0, 1, 3, 2] -> sign = -1
[0, 3, 1, 2] -> sign = 1
[3, 0, 1, 2] -> sign = -1
[3, 0, 2, 1] -> sign = 1
[0, 3, 2, 1] -> sign = -1
[0, 2, 3, 1] -> sign = 1
[0, 2, 1, 3] -> sign = -1
[2, 0, 1, 3] -> sign = 1
[2, 0, 3, 1] -> sign = -1
[2, 3, 0, 1] -> sign = 1
[3, 2, 0, 1] -> sign = -1
[3, 2, 1, 0] -> sign = 1
[2, 3, 1, 0] -> sign = -1
[2, 1, 3, 0] -> sign = 1
[2, 1, 0, 3] -> sign = -1
[1, 2, 0, 3] -> sign = 1
[1, 2, 3, 0] -> sign = -1
[1, 3, 2, 0] -> sign = 1
[3, 1, 2, 0] -> sign = -1
[3, 1, 0, 2] -> sign = 1
[1, 3, 0, 2] -> sign = -1
[1, 0, 3, 2] -> sign = 1
[1, 0, 2, 3] -> sign = -1

Lua

Translation of: C++
_JT={}
function JT(dim)
  local n={ values={}, positions={}, directions={}, sign=1 }
  setmetatable(n,{__index=_JT})
  for i=1,dim do
    n.values[i]=i
    n.positions[i]=i
    n.directions[i]=-1
  end
  return n
end

function _JT:largestMobile()
  for i=#self.values,1,-1 do
    local loc=self.positions[i]+self.directions[i]
    if loc >= 1 and loc <= #self.values and self.values[loc] < i then
      return i
    end
  end
  return 0
end

function _JT:next()
  local r=self:largestMobile()
  if r==0 then return false end
  local rloc=self.positions[r]
  local lloc=rloc+self.directions[r]
  local l=self.values[lloc]
  self.values[lloc],self.values[rloc] = self.values[rloc],self.values[lloc]
  self.positions[l],self.positions[r] = self.positions[r],self.positions[l]
  self.sign=-self.sign
  for i=r+1,#self.directions do self.directions[i]=-self.directions[i] end
  return true
end  

-- test

perm=JT(4)
repeat
  print(unpack(perm.values))
until not perm:next()
Output:
1       2       3       4
1       2       4       3
1       4       2       3
4       1       2       3
4       1       3       2
1       4       3       2
1       3       4       2
1       3       2       4
3       1       2       4
3       1       4       2
3       4       1       2
4       3       1       2
4       3       2       1
3       4       2       1
3       2       4       1
3       2       1       4
2       3       1       4
2       3       4       1
2       4       3       1
4       2       3       1
4       2       1       3
2       4       1       3
2       1       4       3
2       1       3       4

Coroutine Implementation

This is adapted from the Lua Book .

local wrap, yield = coroutine.wrap, coroutine.yield
local function perm(n)
    local r = {}
    for i=1,n do r[i]=i end    
    local sign = 1
  return wrap(function()
    local function swap(m)      
      if m==0 then  
        sign = -sign, yield(sign,r) 
      else
        for i=m,1,-1 do
          r[i],r[m]=r[m],r[i]
          swap(m-1)
          r[i],r[m]=r[m],r[i]
        end    
      end
    end
    swap(n)
  end)
end
for sign,r in perm(3) do print(sign,table.unpack(r))end

Mathematica/Wolfram Language

Recursive

perms[0] = {{{}, 1}}; 
perms[n_] := 
 Flatten[If[#2 == 1, Reverse, # &]@
     Table[{Insert[#1, n, i], (-1)^(n + i) #2}, {i, n}] & @@@ 
   perms[n - 1], 1];

Example:

Print["Perm: ", #[[1]], " Sign: ", #[[2]]] & /@ perms@4;
Output:
Perm: {1,2,3,4} Sign: 1
Perm: {1,2,4,3} Sign: -1
Perm: {1,4,2,3} Sign: 1
Perm: {4,1,2,3} Sign: -1
Perm: {4,1,3,2} Sign: 1
Perm: {1,4,3,2} Sign: -1
Perm: {1,3,4,2} Sign: 1
Perm: {1,3,2,4} Sign: -1
Perm: {3,1,2,4} Sign: 1
Perm: {3,1,4,2} Sign: -1
Perm: {3,4,1,2} Sign: 1
Perm: {4,3,1,2} Sign: -1
Perm: {4,3,2,1} Sign: 1
Perm: {3,4,2,1} Sign: -1
Perm: {3,2,4,1} Sign: 1
Perm: {3,2,1,4} Sign: -1
Perm: {2,3,1,4} Sign: 1
Perm: {2,3,4,1} Sign: -1
Perm: {2,4,3,1} Sign: 1
Perm: {4,2,3,1} Sign: -1
Perm: {4,2,1,3} Sign: 1
Perm: {2,4,1,3} Sign: -1
Perm: {2,1,4,3} Sign: 1
Perm: {2,1,3,4} Sign: -1

Nim

# iterative Boothroyd method
iterator permutations*[T](ys: openarray[T]): tuple[perm: seq[T], sign: int] =
  var
    d = 1
    c = newSeq[int](ys.len)
    xs = newSeq[T](ys.len)
    sign = 1

  for i, y in ys: xs[i] = y
  yield (xs, sign)

  block outter:
    while true:
      while d > 1:
        dec d
        c[d] = 0
      while c[d] >= d:
        inc d
        if d >= ys.len: break outter

      let i = if (d and 1) == 1: c[d] else: 0
      swap xs[i], xs[d]
      sign *= -1
      yield (xs, sign)
      inc c[d]

when isMainModule:
  for i in permutations([0,1,2]):
    echo i

  echo ""

  for i in permutations([0,1,2,3]):
    echo i
Output:
(perm: @[0, 1, 2], sign: 1)
(perm: @[1, 0, 2], sign: -1)
(perm: @[2, 0, 1], sign: 1)
(perm: @[0, 2, 1], sign: -1)
(perm: @[1, 2, 0], sign: 1)
(perm: @[2, 1, 0], sign: -1)

(perm: @[0, 1, 2, 3], sign: 1)
(perm: @[1, 0, 2, 3], sign: -1)
(perm: @[2, 0, 1, 3], sign: 1)
(perm: @[0, 2, 1, 3], sign: -1)
(perm: @[1, 2, 0, 3], sign: 1)
(perm: @[2, 1, 0, 3], sign: -1)
(perm: @[3, 1, 0, 2], sign: 1)
(perm: @[1, 3, 0, 2], sign: -1)
(perm: @[0, 3, 1, 2], sign: 1)
(perm: @[3, 0, 1, 2], sign: -1)
(perm: @[1, 0, 3, 2], sign: 1)
(perm: @[0, 1, 3, 2], sign: -1)
(perm: @[0, 2, 3, 1], sign: 1)
(perm: @[2, 0, 3, 1], sign: -1)
(perm: @[3, 0, 2, 1], sign: 1)
(perm: @[0, 3, 2, 1], sign: -1)
(perm: @[2, 3, 0, 1], sign: 1)
(perm: @[3, 2, 0, 1], sign: -1)
(perm: @[3, 2, 1, 0], sign: 1)
(perm: @[2, 3, 1, 0], sign: -1)
(perm: @[1, 3, 2, 0], sign: 1)
(perm: @[3, 1, 2, 0], sign: -1)
(perm: @[2, 1, 3, 0], sign: 1)
(perm: @[1, 2, 3, 0], sign: -1)

ooRexx

Recursive

/* REXX Compute permutations of things elements       */
/* implementing Heap's algorithm nicely shown in      */
/*   https://en.wikipedia.org/wiki/Heap%27s_algorithm */
/* Recursive Algorithm                                */
Parse Arg things
e.=''
Select
  When things='?' Then
    Call help
  When things='' Then
    things=4
  When words(things)>1 Then Do
    elements=things
    things=words(things)
    Do i=0 By 1 While elements<>''
      Parse Var elements e.i elements
      End
    End
  Otherwise
    If datatype(things)<>'NUM' Then Call help 'bunch ('bunch') must be numeric'
  End
n=0
Do i=0 To things-1
  a.i=i
  End
Call generate things
Say time('R') 'seconds'
Exit

generate: Procedure Expose a. n e. things
  Parse Arg k
  If k=1 Then
    Call show
  Else Do
    Call generate k-1
    Do i=0 To k-2
      ka=k-1
      If k//2=0 Then
        Parse Value a.i a.ka With a.ka a.i
      Else
        Parse Value a.0 a.ka With a.ka a.0
      Call generate k-1
      End
    End
  Return

show: Procedure Expose a. n e. things
  n=n+1
  ol=''
  Do i=0 To things-1
    z=a.i
    If e.0<>'' Then
      ol=ol e.z
    Else
      ol=ol z
    End
  Say strip(ol)
  Return
Exit

help:
  Parse Arg msg
  If msg<>'' Then Do
    Say 'ERROR:' msg
    Say ''
    End
  Say 'rexx permx            -> Permutations of 1 2 3 4               '
  Say 'rexx permx 2          -> Permutations of 1 2                   '
  Say 'rexx permx a b c d    -> Permutations of a b c d in 2 positions'
  Exit
Output:
H:\>rexx permx ?
rexx permx            -> Permutations of 1 2 3 4
rexx permx 2          -> Permutations of 1 2
rexx permx a b c d    -> Permutations of a b c d in 2 positions

H:\>rexx permx 2
0 1
1 0
0 seconds

H:\>rexx permx a b c
a b c
b a c
c a b
a c b
b c a
c b a
0 seconds

Iterative

/* REXX Compute permutations of things elements       */
/* implementing Heap's algorithm nicely shown in      */
/*   https://en.wikipedia.org/wiki/Heap%27s_algorithm */
/* Iterative Algorithm                                */
Parse Arg things
e.=''
Select
  When things='?' Then
    Call help
  When things='' Then
    things=4
  When words(things)>1 Then Do
    elements=things
    things=words(things)
    Do i=0 By 1 While elements<>''
      Parse Var elements e.i elements
      End
    End
  Otherwise
    If datatype(things)<>'NUM' Then Call help 'bunch ('bunch') must be numeric'
  End
Do i=0 To things-1
  a.i=i
  End
Call time 'R'
Call generate things
Say time('E') 'seconds'
Exit

generate:
  Parse Arg n
  Call show
  c.=0
  i=0
  Do While i<n
    If c.i<i Then Do
      if i//2=0 Then
        Parse Value a.0 a.i With a.i a.0
      Else Do
        z=c.i
        Parse Value a.z a.i With a.i a.z
        End
      Call show
      c.i=c.i+1
      i=0
      End
    Else Do
      c.i=0
      i=i+1
      End
    End
  Return

show:
  ol=''
  Do j=0 To n-1
    z=a.j
    If e.0<>'' Then
      ol=ol e.z
    Else
      ol=ol z
    End
  Say strip(ol)
  Return
Exit

help:
  Parse Arg msg
  If msg<>'' Then Do
    Say 'ERROR:' msg
    Say ''
    End
  Say 'rexx permxi            -> Permutations of 1 2 3 4               '
  Say 'rexx permxi 2          -> Permutations of 1 2                   '
  Say 'rexx permxi a b c d    -> Permutations of a b c d in 2 positions'
  Exit

Perl

S-J-T Based

use strict;
use warnings;

# This code uses "Even's Speedup," as described on
# the Wikipedia page about the Steinhaus–Johnson–
# Trotter algorithm.

# Any resemblance between this code and the Python
# code elsewhere on the page is purely a coincidence,
# caused by them both implementing the same algorithm.

# The code was written to be read relatively easily
# while demonstrating some common perl idioms.

sub perms :prototype(&@) {
   my $callback = shift;
   my @perm = map [$_, -1], @_;
   $perm[0][1] = 0;

   my $sign = 1;
   while( ) {
      $callback->($sign, map $_->[0], @perm);
      $sign *= -1;

      my ($chosen, $index) = (-1, -1);
      for my $i ( 0 .. $#perm ) {
         ($chosen, $index) = ($perm[$i][0], $i)
           if $perm[$i][1] and $perm[$i][0] > $chosen;
      }
      return if $index == -1;

      my $direction = $perm[$index][1];
      my $next = $index + $direction;

      @perm[ $index, $next ] = @perm[ $next, $index ];

      if( $next <= 0 or $next >= $#perm ) {
         $perm[$next][1] = 0;
      } elsif( $perm[$next + $direction][0] > $chosen ) {
         $perm[$next][1] = 0;
      }

      for my $i ( 0 .. $next - 1 ) {
         $perm[$i][1] = +1 if $perm[$i][0] > $chosen;
      }
      for my $i ( $next + 1 .. $#perm ) {
         $perm[$i][1] = -1 if $perm[$i][0] > $chosen;
      }
   }
}

my $n = shift(@ARGV) || 4;

perms {
   my ($sign, @perm) = @_;
   print "[", join(", ", @perm), "]";
   print $sign < 0 ? " => -1\n" : " => +1\n";   
} 1 .. $n;
Output:

[1, 2, 3, 4] => +1 [1, 2, 4, 3] => -1 [1, 4, 2, 3] => +1 [4, 1, 2, 3] => -1 [4, 1, 3, 2] => +1 [1, 4, 3, 2] => -1 [1, 3, 4, 2] => +1 [1, 3, 2, 4] => -1 [3, 1, 2, 4] => +1 [3, 1, 4, 2] => -1 [3, 4, 1, 2] => +1 [4, 3, 1, 2] => -1 [4, 3, 2, 1] => +1 [3, 4, 2, 1] => -1 [3, 2, 4, 1] => +1 [3, 2, 1, 4] => -1 [2, 3, 1, 4] => +1 [2, 3, 4, 1] => -1 [2, 4, 3, 1] => +1 [4, 2, 3, 1] => -1 [4, 2, 1, 3] => +1 [2, 4, 1, 3] => -1 [2, 1, 4, 3] => +1 [2, 1, 3, 4] => -1

Alternative Iterative version

This is based on the Raku recursive version, but without recursion.

#!perl
use strict;
use warnings;

sub perms {
   my ($xx) = (shift);
   my @perms = ([+1]);
   for my $x ( 1 .. $xx ) {
      my $sign = -1;
      @perms = map {
         my ($s, @p) = @$_;
         map [$sign *= -1, @p[0..$_-1], $x, @p[$_..$#p]],
            $s < 0 ? 0 .. @p : reverse 0 .. @p;
      } @perms;
   }
   @perms;
}

my $n = shift() || 4;

for( perms($n) ) {
   my $s = shift @$_;
   $s = '+1' if $s > 0;
   print "[", join(", ", @$_), "] => $s\n";
}
Output:

The output is the same as the first perl solution.

Phix

Ad-hoc recursive solution, not (knowingly) based on any given algorithm, but instead on achieving the desired pattern.
Only once finished did I properly grasp that odd/even permutation idea, and that it is very nearly the same algorithm.
Only difference is my version directly calculates where to insert p, without using the parity (which I added in last).

function spermutations(integer p, integer i)
--
-- generate the i'th permutation of [1..p]:
-- first obtain the appropriate permutation of [1..p-1],
-- then insert p/move it down k(=0..p-1) places from the end.
--  
    sequence res
    integer k = mod(i-1,2*p)
    if k>=p then k=2*p-1-k  end if
    if p>1 then
        res = spermutations(p-1,floor((i-1)/p)+1)
        res = res[1..length(res)-k]&p&res[length(res)-k+1..$]
    else
        res = {1}
    end if
    return res
end function
 
for p=1 to 4 do
    printf(1,"==%d==\n",p)
    for i=1 to factorial(p) do
        integer parity = iff(and_bits(i,1)?1:-1)
        ?{i,spermutations(p,i),parity}
    end for
end for
Output:
==1==
{1,{1},1}
==2==
{1,{1,2},1}
{2,{2,1},-1}
==3==
{1,{1,2,3},1}
{2,{1,3,2},-1}
{3,{3,1,2},1}
{4,{3,2,1},-1}
{5,{2,3,1},1}
{6,{2,1,3},-1}
==4==
{1,{1,2,3,4},1}
{2,{1,2,4,3},-1}
{3,{1,4,2,3},1}
{4,{4,1,2,3},-1}
{5,{4,1,3,2},1}
{6,{1,4,3,2},-1}
{7,{1,3,4,2},1}
{8,{1,3,2,4},-1}
{9,{3,1,2,4},1}
{10,{3,1,4,2},-1}
{11,{3,4,1,2},1}
{12,{4,3,1,2},-1}
{13,{4,3,2,1},1}
{14,{3,4,2,1},-1}
{15,{3,2,4,1},1}
{16,{3,2,1,4},-1}
{17,{2,3,1,4},1}
{18,{2,3,4,1},-1}
{19,{2,4,3,1},1}
{20,{4,2,3,1},-1}
{21,{4,2,1,3},1}
{22,{2,4,1,3},-1}
{23,{2,1,4,3},1}
{24,{2,1,3,4},-1}

PicoLisp

(let
   (N 4
      L
      (mapcar
         '((I) (list I 0))
         (range 1 N) ) )
   (for I L
      (printsp (car I)) )
   (prinl)
   (while
      # find the lagest mobile integer
      (setq
         X
         (maxi
            '((I) (car (get L (car I))))
            (extract
               '((I J)
                  (let? Y
                     (get
                        L
                        ((if (=0 (cadr I)) dec inc) J) )
                     (when (> (car I) (car Y))
                        (list J (cadr I)) ) ) )
               L
               (range 1 N) ) )
         Y (get L (car X)) )
      # swap integer and adjacent int it is looking at
      (xchg
         (nth L (car X))
         (nth
            L
            ((if (=0 (cadr X)) dec inc) (car X)) ) )
      # reverse direction of all ints large than our
      (for I L
         (when (< (car Y) (car I))
            (set (cdr I)
               (if (=0 (cadr I)) 1 0) ) ) )
      # print current positions
      (for I L
         (printsp (car I)) )
      (prinl) ) )
(bye)

PowerShell

function output([Object[]]$A, [Int]$k, [ref]$sign) 
{
    "Perm: [$([String]::Join(', ', $A))] Sign: $($sign.Value)"
}

function permutation([Object[]]$array)
{
    function generate([Object[]]$A, [Int]$k, [ref]$sign) 
    {
        if($k -eq 1) 
        {
            output $A $k $sign
            $sign.Value = -$sign.Value
        }
        else
        {
            $k -= 1
            generate $A  $k $sign
            for([Int]$i = 0; $i -lt $k; $i += 1)
             {
                if($i % 2 -eq 0)
                {
                    $A[$i], $A[$k] = $A[$k], $A[$i]
                }
                else
                {
                    $A[0], $A[$k] = $A[$k], $A[0]
                }
                generate $A $k $sign
            }
        }
    }
    generate $array $array.Count ([ref]1)
}
permutation @(0, 1, 2)
""
permutation @(0, 1, 2, 3)

Output:

Perm: [1, 0, 2] Sign: -1
Perm: [2, 0, 1] Sign: 1
Perm: [0, 2, 1] Sign: -1
Perm: [1, 2, 0] Sign: 1
Perm: [2, 1, 0] Sign: -1

Perm: [0, 1, 2, 3] Sign: 1
Perm: [1, 0, 2, 3] Sign: -1
Perm: [2, 0, 1, 3] Sign: 1
Perm: [0, 2, 1, 3] Sign: -1
Perm: [1, 2, 0, 3] Sign: 1
Perm: [2, 1, 0, 3] Sign: -1
Perm: [3, 1, 0, 2] Sign: 1
Perm: [1, 3, 0, 2] Sign: -1
Perm: [0, 3, 1, 2] Sign: 1
Perm: [3, 0, 1, 2] Sign: -1
Perm: [1, 0, 3, 2] Sign: 1
Perm: [0, 1, 3, 2] Sign: -1
Perm: [2, 1, 3, 0] Sign: 1
Perm: [1, 2, 3, 0] Sign: -1
Perm: [3, 2, 1, 0] Sign: 1
Perm: [2, 3, 1, 0] Sign: -1
Perm: [1, 3, 2, 0] Sign: 1
Perm: [3, 1, 2, 0] Sign: -1
Perm: [3, 1, 0, 2] Sign: 1
Perm: [1, 3, 0, 2] Sign: -1
Perm: [0, 3, 1, 2] Sign: 1
Perm: [3, 0, 1, 2] Sign: -1
Perm: [1, 0, 3, 2] Sign: 1
Perm: [0, 1, 3, 2] Sign: -1

Python

Python: iterative

When saved in a file called spermutations.py it is used in the Python example to the Matrix arithmetic task and so any changes here should also be reflected and checked in that task example too.

from operator import itemgetter
 
DEBUG = False # like the built-in __debug__
 
def spermutations(n):
    """permutations by swapping. Yields: perm, sign"""
    sign = 1
    p = [[i, 0 if i == 0 else -1] # [num, direction]
         for i in range(n)]
 
    if DEBUG: print ' #', p
    yield tuple(pp[0] for pp in p), sign
 
    while any(pp[1] for pp in p): # moving
        i1, (n1, d1) = max(((i, pp) for i, pp in enumerate(p) if pp[1]),
                           key=itemgetter(1))
        sign *= -1
        if d1 == -1:
            # Swap down
            i2 = i1 - 1
            p[i1], p[i2] = p[i2], p[i1]
            # If this causes the chosen element to reach the First or last
            # position within the permutation, or if the next element in the
            # same direction is larger than the chosen element:
            if i2 == 0 or p[i2 - 1][0] > n1:
                # The direction of the chosen element is set to zero
                p[i2][1] = 0
        elif d1 == 1:
            # Swap up
            i2 = i1 + 1
            p[i1], p[i2] = p[i2], p[i1]
            # If this causes the chosen element to reach the first or Last
            # position within the permutation, or if the next element in the
            # same direction is larger than the chosen element:
            if i2 == n - 1 or p[i2 + 1][0] > n1:
                # The direction of the chosen element is set to zero
                p[i2][1] = 0
        if DEBUG: print ' #', p
        yield tuple(pp[0] for pp in p), sign
 
        for i3, pp in enumerate(p):
            n3, d3 = pp
            if n3 > n1:
                pp[1] = 1 if i3 < i2 else -1
                if DEBUG: print ' # Set Moving'
 
 
if __name__ == '__main__':
    from itertools import permutations
 
    for n in (3, 4):
        print '\nPermutations and sign of %i items' % n
        sp = set()
        for i in spermutations(n):
            sp.add(i[0])
            print('Perm: %r Sign: %2i' % i)
            #if DEBUG: raw_input('?')
        # Test
        p = set(permutations(range(n)))
        assert sp == p, 'Two methods of generating permutations do not agree'
Output:
Permutations and sign of 3 items
Perm: (0, 1, 2) Sign:  1
Perm: (0, 2, 1) Sign: -1
Perm: (2, 0, 1) Sign:  1
Perm: (2, 1, 0) Sign: -1
Perm: (1, 2, 0) Sign:  1
Perm: (1, 0, 2) Sign: -1

Permutations and sign of 4 items
Perm: (0, 1, 2, 3) Sign:  1
Perm: (0, 1, 3, 2) Sign: -1
Perm: (0, 3, 1, 2) Sign:  1
Perm: (3, 0, 1, 2) Sign: -1
Perm: (3, 0, 2, 1) Sign:  1
Perm: (0, 3, 2, 1) Sign: -1
Perm: (0, 2, 3, 1) Sign:  1
Perm: (0, 2, 1, 3) Sign: -1
Perm: (2, 0, 1, 3) Sign:  1
Perm: (2, 0, 3, 1) Sign: -1
Perm: (2, 3, 0, 1) Sign:  1
Perm: (3, 2, 0, 1) Sign: -1
Perm: (3, 2, 1, 0) Sign:  1
Perm: (2, 3, 1, 0) Sign: -1
Perm: (2, 1, 3, 0) Sign:  1
Perm: (2, 1, 0, 3) Sign: -1
Perm: (1, 2, 0, 3) Sign:  1
Perm: (1, 2, 3, 0) Sign: -1
Perm: (1, 3, 2, 0) Sign:  1
Perm: (3, 1, 2, 0) Sign: -1
Perm: (3, 1, 0, 2) Sign:  1
Perm: (1, 3, 0, 2) Sign: -1
Perm: (1, 0, 3, 2) Sign:  1
Perm: (1, 0, 2, 3) Sign: -1

Python: recursive

After spotting the pattern of highest number being inserted into each perm of lower numbers from right to left, then left to right, I developed this recursive function:

def s_permutations(seq):
    def s_perm(seq):
        if not seq:
            return [[]]
        else:
            new_items = []
            for i, item in enumerate(s_perm(seq[:-1])):
                if i % 2:
                    # step up
                    new_items += [item[:i] + seq[-1:] + item[i:]
                                  for i in range(len(item) + 1)]
                else:
                    # step down
                    new_items += [item[:i] + seq[-1:] + item[i:]
                                  for i in range(len(item), -1, -1)]
            return new_items

    return [(tuple(item), -1 if i % 2 else 1)
            for i, item in enumerate(s_perm(seq))]
Sample output:

The output is the same as before except it is a list of all results rather than yielding each result from a generator function.

Python: Iterative version of the recursive

Replacing the recursion in the example above produces this iterative version function:

def s_permutations(seq):
    items = [[]]
    for j in seq:
        new_items = []
        for i, item in enumerate(items):
            if i % 2:
                # step up
                new_items += [item[:i] + [j] + item[i:]
                              for i in range(len(item) + 1)]
            else:
                # step down
                new_items += [item[:i] + [j] + item[i:]
                              for i in range(len(item), -1, -1)]
        items = new_items

    return [(tuple(item), -1 if i % 2 else 1)
            for i, item in enumerate(items)]
Sample output:

The output is the same as before and is a list of all results rather than yielding each result from a generator function.

Quackery

  [ stack ]                   is parity (   --> s )

  [ 1 & ]                     is odd    ( n --> b )

  [ [] swap witheach
    [ nested
      i odd 2 * 1 -
      join nested join ] ]    is +signs ( [ --> [ )

  [ dup
    [ dup 0 = iff
        [ drop ' [ [ ] ] ]
        done
      dup temp put
      1 - recurse
      [] swap
      witheach
        [ i odd parity put
          temp share times
            [ temp share 1 -
              over
              parity share
              iff i else i^
              stuff
              nested rot join
              swap ]
          drop
          parity release ]
      temp release ]
    swap odd if reverse
    +signs ]                  is perms  ( n --> [ )

  3 perms witheach [ echo cr ]
  cr
  4 perms witheach [ echo cr ]
Output:
[ [ 0 1 2 ] 1 ]
[ [ 0 2 1 ] -1 ]
[ [ 2 0 1 ] 1 ]
[ [ 2 1 0 ] -1 ]
[ [ 1 2 0 ] 1 ]
[ [ 1 0 2 ] -1 ]

[ [ 0 1 2 3 ] 1 ]
[ [ 0 1 3 2 ] -1 ]
[ [ 0 3 1 2 ] 1 ]
[ [ 3 0 1 2 ] -1 ]
[ [ 3 0 2 1 ] 1 ]
[ [ 0 3 2 1 ] -1 ]
[ [ 0 2 3 1 ] 1 ]
[ [ 0 2 1 3 ] -1 ]
[ [ 2 0 1 3 ] 1 ]
[ [ 2 0 3 1 ] -1 ]
[ [ 2 3 0 1 ] 1 ]
[ [ 3 2 0 1 ] -1 ]
[ [ 3 2 1 0 ] 1 ]
[ [ 2 3 1 0 ] -1 ]
[ [ 2 1 3 0 ] 1 ]
[ [ 2 1 0 3 ] -1 ]
[ [ 1 2 0 3 ] 1 ]
[ [ 1 2 3 0 ] -1 ]
[ [ 1 3 2 0 ] 1 ]
[ [ 3 1 2 0 ] -1 ]
[ [ 3 1 0 2 ] 1 ]
[ [ 1 3 0 2 ] -1 ]
[ [ 1 0 3 2 ] 1 ]
[ [ 1 0 2 3 ] -1 ]

Racket

#lang racket

(define (add-at l i x)
  (if (zero? i) (cons x l) (cons (car l) (add-at (cdr l) (sub1 i) x))))

(define (permutations l)
  (define (loop l)
    (cond [(null? l) '(())]
          [else (for*/list ([(p i) (in-indexed (loop (cdr l)))]
                            [i ((if (odd? i) identity reverse)
                                (range (add1 (length p))))])
                  (add-at p i (car l)))]))
  (for/list ([p (loop (reverse l))] [i (in-cycle '(1 -1))]) (cons i p)))

(define (show-permutations l)
  (printf "Permutations of ~s:\n" l)
  (for ([p (permutations l)])
    (printf "  ~a (~a)\n" (apply ~a (add-between (cdr p) ", ")) (car p))))

(for ([n (in-range 3 5)]) (show-permutations (range n)))
Output:
Permutations of (0 1 2):
  0, 1, 2 (1)
  0, 2, 1 (-1)
  2, 0, 1 (1)
  2, 1, 0 (-1)
  1, 2, 0 (1)
  1, 0, 2 (-1)
Permutations of (0 1 2 3):
  0, 1, 2, 3 (1)
  0, 1, 3, 2 (-1)
  0, 3, 1, 2 (1)
  3, 0, 1, 2 (-1)
  3, 0, 2, 1 (1)
  0, 3, 2, 1 (-1)
  0, 2, 3, 1 (1)
  0, 2, 1, 3 (-1)
  2, 0, 1, 3 (1)
  2, 0, 3, 1 (-1)
  2, 3, 0, 1 (1)
  3, 2, 0, 1 (-1)
  3, 2, 1, 0 (1)
  2, 3, 1, 0 (-1)
  2, 1, 3, 0 (1)
  2, 1, 0, 3 (-1)
  1, 2, 0, 3 (1)
  1, 2, 3, 0 (-1)
  1, 3, 2, 0 (1)
  3, 1, 2, 0 (-1)
  3, 1, 0, 2 (1)
  1, 3, 0, 2 (-1)
  1, 0, 3, 2 (1)
  1, 0, 2, 3 (-1)

Raku

(formerly Perl 6)

Recursive

Works with: rakudo version 2015-09-25
sub insert($x, @xs) { ([flat @xs[0 ..^ $_], $x, @xs[$_ .. *]] for 0 .. +@xs) }
sub order($sg, @xs) { $sg > 0 ?? @xs !! @xs.reverse }
 
multi perms([]) {
    [] => +1
}
 
multi perms([$x, *@xs]) {
    perms(@xs).map({ |order($_.value, insert($x, $_.key)) }) Z=> |(+1,-1) xx *
}
 
.say for perms([0..2]);
Output:
[0 1 2] => 1
[1 0 2] => -1
[1 2 0] => 1
[2 1 0] => -1
[2 0 1] => 1
[0 2 1] => -1

REXX

Version 1

This program does not work asdescribed in the comment section and I can't get it working for 5 things. -:( --Walter Pachl 13:40, 25 January 2022 (UTC)

/*REXX program  generates all  permutations  of   N   different objects by  swapping.   */
parse arg things bunch .                         /*obtain optional arguments from the CL*/
if things=='' | things==","  then things=4       /*Not specified?  Then use the default.*/
if bunch =='' | bunch ==","  then bunch =things  /* "      "         "   "   "     "    */
call permSets things, bunch                      /*invoke permutations by swapping sub. */
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
!:        procedure;  !=1;        do j=2  to arg(1);    !=!*j;     end;           return !
/*──────────────────────────────────────────────────────────────────────────────────────*/
permSets: procedure; parse arg x,y               /*take   X  things   Y   at a time.    */
          !.=0;      pad=left('', x*y)           /*X can't be > length of below str (62)*/
          z=left('123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ', x);  q=z
          #=1                                    /*the number of permutations  (so far).*/
          !.z=1;    s=1;   times=!(x) % !(x-y)   /*calculate (#) TIMES  using factorial.*/
          w=max(length(z), length('permute') )   /*maximum width of  Z and also PERMUTE.*/
          say center('permutations for '   x   ' things taken '   y   " at a time",60,'═')
          say
          say   pad    'permutation'       center("permute", w, '─')         "sign"
          say   pad    '───────────'       center("───────", w, '─')         "────"
          say   pad    center(#, 11)       center(z        , w)              right(s, 4-1)

             do $=1   until  #==times            /*perform permutation until # of times.*/
               do   k=1    for x-1               /*step thru things for  things-1 times.*/
                 do m=k+1  to  x;      ?=        /*this method doesn't use  adjacency.  */
                     do n=1  for x               /*build the new permutation by swapping*/
                     if n\==k & n\==m  then               ? =  ?  ||  substr(z, n, 1)
                                       else if n==k  then ? =  ?  ||  substr(z, m, 1)
                                                     else ? =  ?  ||  substr(z, k, 1)
                     end   /*n*/
                 z=?                             /*save this permutation for next swap. */
                 if !.?  then iterate m          /*if defined before, then try next one.*/
                 _=0                             /* [↓]  count number of swapped symbols*/
                    do d=1  for x  while $\==1;  _= _ + (substr(?,d,1)\==substr(prev,d,1))
                    end   /*d*/
                 if _>2  then do;        _=z
                              a=$//x+1;  q=q + _ /* [← ↓]  this swapping tries adjacency*/
                              b=q//x+1;  if b==a  then b=a + 1;       if b>x  then b=a - 1
                              z=overlay( substr(z,b,1), overlay( substr(z,a,1), _, b),  a)
                              iterate $          /*now, try this particular permutation.*/
                              end
                 #=#+1;  s= -s;   say pad   center(#, 11)    center(?, w)    right(s, 4-1)
                 !.?=1;  prev=?;      iterate $  /*now, try another swapped permutation.*/
                 end   /*m*/
               end     /*k*/
             end       /*$*/
          return                                 /*we're all finished with permutating. */
output   when using the default input:
══════permutations for  4  things taken  4  at a time═══════

                 permutation permute sign
                 ─────────── ─────── ────
                      1       1234     1
                      2       2134    -1
                      3       3124     1
                      4       1324    -1
                      5       1342     1
                      6       3142    -1
                      7       4132     1
                      8       1432    -1
                      9       2431     1
                     10       4231    -1
                     11       4321     1
                     12       3421    -1
                     13       3241     1
                     14       2341    -1
                     15       2314     1
                     16       3214    -1
                     17       3412     1
                     18       4312    -1
                     19       4213     1
                     20       2413    -1
                     21       2143     1
                     22       1243    -1
                     23       1423     1
                     24       4123    -1

Version 2

See program shown for ooRexx

Ruby

Translation of: BBC BASIC
def perms(n)
  p = Array.new(n+1){|i| -i}
  s = 1
  loop do
    yield p[1..-1].map(&:abs), s
    k = 0
    for i in 2..n
      k = i if p[i] < 0 and p[i].abs > p[i-1].abs and p[i].abs > p[k].abs
    end
    for i in 1...n
      k = i if p[i] > 0 and p[i].abs > p[i+1].abs and  p[i].abs > p[k].abs
    end
    break if k.zero?
    for i in 1..n
      p[i] *= -1 if p[i].abs > p[k].abs
    end
    i = k + (p[k] <=> 0)
    p[k], p[i] = p[i], p[k]
    s = -s
  end
end

for i in 3..4
  perms(i){|perm, sign| puts "Perm: #{perm}  Sign: #{sign}"}
  puts
end
Output:
Perm: [1, 2, 3]  Sign: 1
Perm: [1, 3, 2]  Sign: -1
Perm: [3, 1, 2]  Sign: 1
Perm: [3, 2, 1]  Sign: -1
Perm: [2, 3, 1]  Sign: 1
Perm: [2, 1, 3]  Sign: -1

Perm: [1, 2, 3, 4]  Sign: 1
Perm: [1, 2, 4, 3]  Sign: -1
Perm: [1, 4, 2, 3]  Sign: 1
Perm: [4, 1, 2, 3]  Sign: -1
Perm: [4, 1, 3, 2]  Sign: 1
Perm: [1, 4, 3, 2]  Sign: -1
Perm: [1, 3, 4, 2]  Sign: 1
Perm: [1, 3, 2, 4]  Sign: -1
Perm: [3, 1, 2, 4]  Sign: 1
Perm: [3, 1, 4, 2]  Sign: -1
Perm: [3, 4, 1, 2]  Sign: 1
Perm: [4, 3, 1, 2]  Sign: -1
Perm: [4, 3, 2, 1]  Sign: 1
Perm: [3, 4, 2, 1]  Sign: -1
Perm: [3, 2, 4, 1]  Sign: 1
Perm: [3, 2, 1, 4]  Sign: -1
Perm: [2, 3, 1, 4]  Sign: 1
Perm: [2, 3, 4, 1]  Sign: -1
Perm: [2, 4, 3, 1]  Sign: 1
Perm: [4, 2, 3, 1]  Sign: -1
Perm: [4, 2, 1, 3]  Sign: 1
Perm: [2, 4, 1, 3]  Sign: -1
Perm: [2, 1, 4, 3]  Sign: 1
Perm: [2, 1, 3, 4]  Sign: -1

Rust

// Implementation of Heap's algorithm.
// See https://en.wikipedia.org/wiki/Heap%27s_algorithm#Details_of_the_algorithm
fn generate<T, F>(a: &mut [T], output: F)
where
    F: Fn(&[T], isize),
{
    let n = a.len();
    let mut c = vec![0; n];
    let mut i = 1;
    let mut sign = 1;
    output(a, sign);
    while i < n {
        if c[i] < i {
            if (i & 1) == 0 {
                a.swap(0, i);
            } else {
                a.swap(c[i], i);
            }
            sign = -sign;
            output(a, sign);
            c[i] += 1;
            i = 1;
        } else {
            c[i] = 0;
            i += 1;
        }
    }
}

fn print_permutation<T: std::fmt::Debug>(a: &[T], sign: isize) {
    println!("{:?} {}", a, sign);
}

fn main() {
    println!("Permutations and signs for three items:");
    let mut a = vec![0, 1, 2];
    generate(&mut a, print_permutation);

    println!("\nPermutations and signs for four items:");
    let mut b = vec![0, 1, 2, 3];
    generate(&mut b, print_permutation);
}
Output:
[0, 1, 2] 1
[1, 0, 2] -1
[2, 0, 1] 1
[0, 2, 1] -1
[1, 2, 0] 1
[2, 1, 0] -1

Permutations and signs for four items:
[0, 1, 2, 3] 1
[1, 0, 2, 3] -1
[2, 0, 1, 3] 1
[0, 2, 1, 3] -1
[1, 2, 0, 3] 1
[2, 1, 0, 3] -1
[3, 1, 0, 2] 1
[1, 3, 0, 2] -1
[0, 3, 1, 2] 1
[3, 0, 1, 2] -1
[1, 0, 3, 2] 1
[0, 1, 3, 2] -1
[0, 2, 3, 1] 1
[2, 0, 3, 1] -1
[3, 0, 2, 1] 1
[0, 3, 2, 1] -1
[2, 3, 0, 1] 1
[3, 2, 0, 1] -1
[3, 2, 1, 0] 1
[2, 3, 1, 0] -1
[1, 3, 2, 0] 1
[3, 1, 2, 0] -1
[2, 1, 3, 0] 1
[1, 2, 3, 0] -1

Scala

object JohnsonTrotter extends App {

  private def perm(n: Int): Unit = {
    val p = new Array[Int](n) // permutation
    val pi = new Array[Int](n) // inverse permutation
    val dir = new Array[Int](n) // direction = +1 or -1

    def perm(n: Int, p: Array[Int], pi: Array[Int], dir: Array[Int]): Unit = {
      if (n >= p.length) for (aP <- p) print(aP)
      else {
        perm(n + 1, p, pi, dir)
        for (i <- 0 until n) { // swap
          printf("   (%d %d)\n", pi(n), pi(n) + dir(n))
          val z = p(pi(n) + dir(n))
          p(pi(n)) = z
          p(pi(n) + dir(n)) = n
          pi(z) = pi(n)
          pi(n) = pi(n) + dir(n)
          perm(n + 1, p, pi, dir)
        }
        dir(n) = -dir(n)
      }
    }

    for (i <- 0 until n) {
      dir(i) = -1
      p(i) = i
      pi(i) = i
    }
    perm(0, p, pi, dir)
    print("   (0 1)\n")
  }

  perm(4)

}
Output:

See it in running in your browser by Scastie (JVM).

Sidef

Translation of: Perl
func perms(n) {
   var perms = [[+1]]
   for x in (1..n) {
      var sign = -1
      perms = gather {
        for s,*p in perms {
          var r = (0 .. p.len)
          take((s < 0 ? r : r.flip).map {|i|
            [sign *= -1, p[^i], x, p[i..p.end]]
          }...)
        }
      }
   }
   perms
}

var n = 4
for p in perms(n) {
    var s = p.shift
    s > 0 && (s = '+1')
    say "#{p} => #{s}"
}
Output:
[1, 2, 3, 4] => +1
[1, 2, 4, 3] => -1
[1, 4, 2, 3] => +1
[4, 1, 2, 3] => -1
[4, 1, 3, 2] => +1
[1, 4, 3, 2] => -1
[1, 3, 4, 2] => +1
[1, 3, 2, 4] => -1
[3, 1, 2, 4] => +1
[3, 1, 4, 2] => -1
[3, 4, 1, 2] => +1
[4, 3, 1, 2] => -1
[4, 3, 2, 1] => +1
[3, 4, 2, 1] => -1
[3, 2, 4, 1] => +1
[3, 2, 1, 4] => -1
[2, 3, 1, 4] => +1
[2, 3, 4, 1] => -1
[2, 4, 3, 1] => +1
[4, 2, 3, 1] => -1
[4, 2, 1, 3] => +1
[2, 4, 1, 3] => -1
[2, 1, 4, 3] => +1
[2, 1, 3, 4] => -1

Swift

// Implementation of Heap's algorithm.
// See https://en.wikipedia.org/wiki/Heap%27s_algorithm#Details_of_the_algorithm
func generate<T>(array: inout [T], output: (_: [T], _: Int) -> Void) {
    let n = array.count
    var c = Array(repeating: 0, count: n)
    var i = 1
    var sign = 1
    output(array, sign)
    while i < n {
        if c[i] < i {
            if (i & 1) == 0 {
                array.swapAt(0, i)
            } else {
                array.swapAt(c[i], i)
            }
            sign = -sign
            output(array, sign)
            c[i] += 1
            i = 1
        } else {
            c[i] = 0
            i += 1
        }
    }
}

func printPermutation<T>(array: [T], sign: Int) {
    print("\(array) \(sign)")
}

print("Permutations and signs for three items:")
var a = [0, 1, 2]
generate(array: &a, output: printPermutation)

print("\nPermutations and signs for four items:")
var b = [0, 1, 2, 3]
generate(array: &b, output: printPermutation)
Output:
Permutations and signs for three items:
[0, 1, 2] 1
[1, 0, 2] -1
[2, 0, 1] 1
[0, 2, 1] -1
[1, 2, 0] 1
[2, 1, 0] -1

Permutations and signs for four items:
[0, 1, 2, 3] 1
[1, 0, 2, 3] -1
[2, 0, 1, 3] 1
[0, 2, 1, 3] -1
[1, 2, 0, 3] 1
[2, 1, 0, 3] -1
[3, 1, 0, 2] 1
[1, 3, 0, 2] -1
[0, 3, 1, 2] 1
[3, 0, 1, 2] -1
[1, 0, 3, 2] 1
[0, 1, 3, 2] -1
[0, 2, 3, 1] 1
[2, 0, 3, 1] -1
[3, 0, 2, 1] 1
[0, 3, 2, 1] -1
[2, 3, 0, 1] 1
[3, 2, 0, 1] -1
[3, 2, 1, 0] 1
[2, 3, 1, 0] -1
[1, 3, 2, 0] 1
[3, 1, 2, 0] -1
[2, 1, 3, 0] 1
[1, 2, 3, 0] -1

Tcl

# A simple swap operation
proc swap {listvar i1 i2} {
    upvar 1 $listvar l
    set tmp [lindex $l $i1]
    lset l $i1 [lindex $l $i2]
    lset l $i2 $tmp
}

proc permswap {n v1 v2 body} {
    upvar 1 $v1 perm $v2 sign

    # Initialize
    set sign -1
    for {set i 0} {$i < $n} {incr i} {
	lappend items $i
	lappend dirs -1
    }

    while 1 {
	# Report via callback
	set perm $items
	set sign [expr {-$sign}]
	uplevel 1 $body

	# Find the largest mobile integer (lmi) and its index (idx)
	set i [set idx -1]
	foreach item $items dir $dirs {
	    set j [expr {[incr i] + $dir}]
	    if {$j < 0 || $j >= [llength $items]} continue
	    if {$item > [lindex $items $j] && ($idx == -1 || $item > $lmi)} {
		set lmi $item
		set idx $i
	    }
	}

	# If none, we're done
	if {$idx == -1} break

	# Swap the largest mobile integer with "what it is looking at"
	set nextIdx [expr {$idx + [lindex $dirs $idx]}]
	swap items $idx $nextIdx
	swap dirs $idx $nextIdx

	# Reverse directions on larger integers
	set i -1
	foreach item $items dir $dirs {
	    lset dirs [incr i] [expr {$item > $lmi ? -$dir : $dir}]
	}
    }
}

Demonstrating:

permswap 4 p s {
    puts "$s\t$p"
}
Output:
1	0 1 2 3
-1	0 1 3 2
1	0 3 1 2
-1	3 0 1 2
1	3 0 2 1
-1	0 3 2 1
1	0 2 3 1
-1	0 2 1 3
1	2 0 1 3
-1	2 0 3 1
1	2 3 0 1
-1	3 2 0 1
1	3 2 1 0
-1	2 3 1 0
1	2 1 3 0
-1	2 1 0 3
1	1 2 0 3
-1	1 2 3 0
1	1 3 2 0
-1	3 1 2 0
1	3 1 0 2
-1	1 3 0 2
1	1 0 3 2
-1	1 0 2 3

Wren

Translation of: Kotlin
var johnsonTrotter = Fn.new { |n|
    var p = List.filled(n, 0)  // permutation
    var q = List.filled(n, 0)  // inverse permutation
    for (i in 0...n) p[i] = q[i] = i
    var d = List.filled(n, -1) // direction = 1 or -1
    var sign = 1
    var perms = []
    var signs = []

    var permute // recursive closure
    permute = Fn.new { |k|
        if (k >= n) {
            perms.add(p.toList)
            signs.add(sign)
            sign = sign * -1
            return
        }
        permute.call(k + 1)
        for (i in 0...k) {
            var z = p[q[k] + d[k]]
            p[q[k]] = z
            p[q[k] + d[k]] = k
            q[z] = q[k]
            q[k] = q[k] + d[k]
            permute.call(k + 1)
        }
        d[k] = d[k] * -1
    }
    permute.call(0)
    return [perms, signs]
}

var printPermsAndSigns = Fn.new { |perms, signs|
    var i = 0
    for (perm in perms) {
        System.print("%(perm) -> sign = %(signs[i])")
        i = i + 1
    }
}

var res = johnsonTrotter.call(3)
var perms = res[0]
var signs = res[1]
printPermsAndSigns.call(perms, signs)
System.print()
res = johnsonTrotter.call(4)
perms = res[0]
signs = res[1]
printPermsAndSigns.call(perms, signs)
Output:
[0, 1, 2] -> sign = 1
[0, 2, 1] -> sign = -1
[2, 0, 1] -> sign = 1
[2, 1, 0] -> sign = -1
[1, 2, 0] -> sign = 1
[1, 0, 2] -> sign = -1

[0, 1, 2, 3] -> sign = 1
[0, 1, 3, 2] -> sign = -1
[0, 3, 1, 2] -> sign = 1
[3, 0, 1, 2] -> sign = -1
[3, 0, 2, 1] -> sign = 1
[0, 3, 2, 1] -> sign = -1
[0, 2, 3, 1] -> sign = 1
[0, 2, 1, 3] -> sign = -1
[2, 0, 1, 3] -> sign = 1
[2, 0, 3, 1] -> sign = -1
[2, 3, 0, 1] -> sign = 1
[3, 2, 0, 1] -> sign = -1
[3, 2, 1, 0] -> sign = 1
[2, 3, 1, 0] -> sign = -1
[2, 1, 3, 0] -> sign = 1
[2, 1, 0, 3] -> sign = -1
[1, 2, 0, 3] -> sign = 1
[1, 2, 3, 0] -> sign = -1
[1, 3, 2, 0] -> sign = 1
[3, 1, 2, 0] -> sign = -1
[3, 1, 0, 2] -> sign = 1
[1, 3, 0, 2] -> sign = -1
[1, 0, 3, 2] -> sign = 1
[1, 0, 2, 3] -> sign = -1

XPL0

Translation of BBC BASIC example, which uses the Johnson-Trotter algorithm.

include c:\cxpl\codes;

proc PERMS(N);
int  N;                         \number of elements
int  I, K, S, T, P;
[P:= Reserve((N+1)*4);
for I:= 0 to N do P(I):= -I;    \initialize facing left (also set P(0)=0)
S:= 1;
repeat  Text(0, "Perm: [ ");
        for I:= 1 to N do
                [IntOut(0, abs(P(I)));  ChOut(0, ^ )];
        Text(0, "] Sign: ");  IntOut(0, S);  CrLf(0);

        K:= 0;                  \find largest mobile element
        for I:= 2 to N do                         \for left-facing elements
            if P(I) < 0 and
                abs(P(I)) > abs(P(I-1)) and       \ greater than neighbor
                abs(P(I)) > abs(P(K)) then K:= I; \ get largest element
        for I:= 1 to N-1 do                       \for right-facing elements
            if P(I) > 0 and
                abs(P(I)) > abs(P(I+1)) and       \ greater than neighbor
                abs(P(I)) > abs(P(K)) then K:= I; \ get largest element
        if K # 0 then           \mobile element found
           [for I:= 1 to N do   \reverse elements > K
                if abs(P(I)) > abs(P(K)) then P(I):= P(I)*-1;
            I:= K + (if P(K)<0 then -1 else 1);
            T:= P(K);  P(K):= P(I);  P(I):= T;    \swap K with element looked at
            S:= -S;             \alternate signs
            ];
until   K = 0;                  \no mobile element remains
];

[PERMS(3);
CrLf(0);
PERMS(4);
]
Output:
Perm: [ 1 2 3 ] Sign: 1
Perm: [ 1 3 2 ] Sign: -1
Perm: [ 3 1 2 ] Sign: 1
Perm: [ 3 2 1 ] Sign: -1
Perm: [ 2 3 1 ] Sign: 1
Perm: [ 2 1 3 ] Sign: -1

Perm: [ 1 2 3 4 ] Sign: 1
Perm: [ 1 2 4 3 ] Sign: -1
Perm: [ 1 4 2 3 ] Sign: 1
Perm: [ 4 1 2 3 ] Sign: -1
Perm: [ 4 1 3 2 ] Sign: 1
Perm: [ 1 4 3 2 ] Sign: -1
Perm: [ 1 3 4 2 ] Sign: 1
Perm: [ 1 3 2 4 ] Sign: -1
Perm: [ 3 1 2 4 ] Sign: 1
Perm: [ 3 1 4 2 ] Sign: -1
Perm: [ 3 4 1 2 ] Sign: 1
Perm: [ 4 3 1 2 ] Sign: -1
Perm: [ 4 3 2 1 ] Sign: 1
Perm: [ 3 4 2 1 ] Sign: -1
Perm: [ 3 2 4 1 ] Sign: 1
Perm: [ 3 2 1 4 ] Sign: -1
Perm: [ 2 3 1 4 ] Sign: 1
Perm: [ 2 3 4 1 ] Sign: -1
Perm: [ 2 4 3 1 ] Sign: 1
Perm: [ 4 2 3 1 ] Sign: -1
Perm: [ 4 2 1 3 ] Sign: 1
Perm: [ 2 4 1 3 ] Sign: -1
Perm: [ 2 1 4 3 ] Sign: 1
Perm: [ 2 1 3 4 ] Sign: -1

zkl

Translation of: Python
Translation of: Haskell
fcn permute(seq)
{
   insertEverywhere := fcn(x,list){ //(x,(a,b))-->((x,a,b),(a,x,b),(a,b,x))
      (0).pump(list.len()+1,List,'wrap(n){list[0,n].extend(x,list[n,*]) })};
   insertEverywhereB := fcn(x,t){ //--> insertEverywhere().reverse()
      [t.len()..-1,-1].pump(t.len()+1,List,'wrap(n){t[0,n].extend(x,t[n,*])})};

   seq.reduce('wrap(items,x){
      f := Utils.Helpers.cycle(insertEverywhereB,insertEverywhere);
      items.pump(List,'wrap(item){f.next()(x,item)},
	      T.fp(Void.Write,Void.Write));
   },T(T));
}

A cycle of two "build list" functions is used to insert x forward or reverse. reduce loops over the items and retains the enlarging list of permuations. pump loops over the existing set of permutations and inserts/builds the next set (into a list sink). (Void.Write,Void.Write,list) is a sentinel that says to write the contents of the list to the sink (ie sink.extend(list)). T.fp is a partial application of ROList.create (read only list) and the parameters VW,VW. It will be called (by pump) with a list of lists --> T.create(VM,VM,list) --> list

p := permute(T(1,2,3));
p.println();

p := permute([1..4]);
p.len().println();
p.toString(*).println()
Output:
L(L(1,2,3),L(1,3,2),L(3,1,2),L(3,2,1),L(2,3,1),L(2,1,3))

24
L(
L(1,2,3,4), L(1,2,4,3), L(1,4,2,3), L(4,1,2,3), L(4,1,3,2), L(1,4,3,2),
L(1,3,4,2), L(1,3,2,4), L(3,1,2,4), L(3,1,4,2), L(3,4,1,2), L(4,3,1,2), 
L(4,3,2,1), L(3,4,2,1), L(3,2,4,1), L(3,2,1,4), L(2,3,1,4), L(2,3,4,1), 
L(2,4,3,1), L(4,2,3,1), L(4,2,1,3), L(2,4,1,3), L(2,1,4,3), L(2,1,3,4) )

An iterative, lazy version, which is handy as the number of permutations is n!. Uses "Even's Speedup" as described in the Wikipedia article:

 fcn [private] _permuteW(seq){	// lazy version
   N:=seq.len(); NM1:=N-1;
   ds:=(0).pump(N,List,T(Void,-1)).copy(); ds[0]=0; // direction to move e: -1,0,1
   es:=(0).pump(N,List).copy();  // enumerate seq

   while(1) {
      vm.yield(es.pump(List,seq.__sGet));

      // find biggest e with d!=0
      reg i=Void, c=-1;
      foreach n in (N){ if(ds[n] and es[n]>c) { c=es[n]; i=n; } }
      if(Void==i) return();

      d:=ds[i]; j:=i+d;
      es.swap(i,j); ds.swap(i,j);	// d tracks e
      if(j==NM1 or j==0 or es[j+d]>c) ds[j]=0;
      foreach e in (N){ if(es[e]>c) ds[e]=(i-e).sign }
   } 
} 

fcn permuteW(seq) { Utils.Generator(_permuteW,seq) }
foreach p in (permuteW(T("a","b","c"))){ println(p) }
Output:
L("a","b","c")
L("a","c","b")
L("c","a","b")
L("c","b","a")
L("b","c","a")
L("b","a","c")