How to improve efficiency of algorithm which generates next lexicographic permutation?

Question

It must be noted here that I performed the mathematics by hand on paper to derive the foregoing proofs. I am not sure if the proofs would have become apparent by solely usin

Answer 1

Lack of skills in maths and english makes it hard for me to elaborate on such a question :-| Though I wanted to play with permutations and I did something not so off-topic after all. See TL;DR at https://stackoverflow.com/a/43302308/1636522, I have the same feeling and I'll try to demonstrate the validity of my algorithm later, I need a little time to think of it.

var a = "aceg".split(""); 
do {
  console.log(a.join(""));
} while (a = next(a));

function next (a) {
  var i, c, head, next;
  var b = a.slice();
  var n = b.length;
  for (i = 1; i < n; i++) {
    if (b[n - i] > b[n - (i + 1)]) {
      head = n - (i + 1);
      next = n - i;
      for (i = next + 1; i < n; i++) {
        if (b[i] < b[next] && b[i] > b[head]) {
          next = i;
        }
      }
      c = b[head];
      b[head] = b[next];
      b[next] = c;
      return b.slice(0, head + 1).concat(
        b.slice(head + 1).sort()
      );
    }
  }
}