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

 Steinhaus–Johnson–Trotter algorithm
 Johnson-Trotter Algorithm Listing All Permutations
 Heap's algorithm
 [1] Tintinnalogia


Related tasks

   Matrix arithmetic
   Gray code


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}"
}