123
456
789

123 -> 312
456    456
789    789

0 1 2 3  -->  0
0 1 3 2  -->  1
0 2 1 3  -->  2
...
1 0 2 3  -->  6
1 0 3 2  -->  7
...
3 1 2 0  --> 21
3 2 0 1  --> 22
3 2 1 0  --> 23

n = a_1 * 1! + a_2 * 2! + a_3 * 3! + a_4 * 4! + ...

for i = 1 to s-1
    the i-th symbol becomes the (a_(s-i)+1)-th smallest unused symbol
 the last symbol is the left over one

4231 = 1 + 2*2115 :  a_1 = 1
2115 = 0 + 3* 705 :  a_2 = 0
 705 = 1 + 4* 176 :  a_3 = 1
 176 = 1 + 5*  35 :  a_4 = 1
  35 = 5 + 6*   5 :  a_5 = 5
   5 = 5 + 7*   0 :  a_6 = 5

coefficients      symbols       choice
0,5,5,1,1,0,1  1,2,3,4,5,6,7,8     1
 5,5,1,1,0,1    2,3,4,5,6,7,8      7
  5,1,1,0,1      2,3,4,5,6,8       8
   1,1,0,1        2,3,4,5,6        3
    1,0,1          2,4,5,6         4
     0,1            2,5,6          2
      1              5,6           6
      -               5            5

symbols     count     perm    index(head)
1,2,3,4,5,6   6   2,4,6,3,5,1    1         a_5 = 1
 1,3,4,5,6    5    4,6,3,5,1     2         a_4 = 2
  1,3,5,6     4     6,3,5,1      3         a_3 = 3
   1,3,5      3      3,5,1       1         a_2 = 1
    1,5       2       5,1        1         a_1 = 1